Campbell's 15th book does contain a dance with this name, but I have been unable to find more than a table of contents and can't verify the choreography.

Campbell did publish a dance with this title in his 19th Book, but I do not know whether its figure matches that of the RSCDS dance.

On the other hand, in Thompson's Compleat Collection of 200 Fashionable Country Dances, Vol. III", ~1773, Charles and Samuel Thompson write:

Cast off two Cu. and clap hands cast up and do the same cross over two Cu. lead up to the top and clap hands lead down two Cu. up again and cast off and clap hands

Wilson writes (on page 40):

SINGLE FIGURE (Tune played straight thro')
Hey on your own sides lead down the middle up again & set to top Cu:

OR THUS Set & change sides with the 2^d. Cu: set & back again lead down the middle up again & right & left with the top Cu:

DOUBLE FIGURE (Each strain repeated)
The 3 ladies lead round the 3 gent: the 3 gent: lead round the 3 ladies   lead down the middle up again allemande & set 3 across & set 3 in your places  

Wilson usually gives multiple figures for a tune, the (R)SCDS appears to have conflated the first figure and the last. They seem to have taken the first 8 bars from the first figure (the hey), and added the last 16 bars of the final figure, except these have been reinterpreted to take up 24 bars.

The music consists of two 8 bar strains.

Wilson defines most of his figures in his An Analysis of Country Dancing.
Hey on your own sides is only defined as part of "Hey Contrary Sides, and Hey of your own Sides" on page 83 and he shows a mirror/reflection hey.
The 3 ladies lead round the 3 gent: may be found on pages 48-49 and means: "all the ladies take hands, lady 1 leads counter-clockwise above, and then behind the men, at bottom looping back to original places."
lead down the middle up again may be found on pages 10-11 and means: "lead down two couples, turn, lead up to the place of the couple who were below (who move up)" Wilson does not specify a hand hold for the lead, nor whether the 2s take hands.
allemande may be found on page 7 and appears to mean going about your partner in a circle. In a later edition of Analysis Wilson adds the phrase "back to back".
This is an unusual meaning for "allemande", but, as best as I can understand, that is what Wilson describes.
set 3 across & set 3 in your places is only described as "set three in their places and set three across" (opposite order) and is found on pages 76-77, and means: "Two lines of three across set, the ones move to their own places, everyone sets again." Wilson does not say whether they take hands when setting. Wilson does not indicate whether the corners set when the 1s move.

A "bawbee" is not, as I had assumed, a baby, but according to Wikipedia it is a small coin, a Scottish sixpence, worth about an English half-penny.

Wilson writes (on page 67):

SINGLE FIGURE (Each strain repeated)
Set & half right and left with 2^d.Cu: set & back again   lead down the middle up again & allemande  
OR THUS Hands 6 quite round & back again   lead down the middle up again & set to top Cu:  
DOUBLE FIGURE (Tune played twice thro' with repeats)
Set & change sides with 2^d.Cu: set & back again   promenade 3 Cu:   whole poussette   & whole figure contary corners  

Wilson usually gives multiple figures for each tune, the RSCDS appears to be following the first SINGLE FIGURE.

The music consists of two 4 bar strains. I assume the music should be played at 2 beats per bar and that " " indicates two repetitions of each strain.

Wilson defines all his figures in his An Analysis of Country Dancing.
right and left may be found on pages 39-44 and does not mean what I expect, instead: "2nd corners change by left shoulders, 1st corners right, 2nd corners change back by left, 1st corners by right."
lead down the middle up again may be found on pages 10-11 and means: "lead down two couples, turn, lead up to the place of the couple who were below (who move up)" Wilson does not specify a hand hold for the lead, nor whether the 2s take hands.
allemande may be found on page 7 and appears to mean going about your partner in a circle. In a later edition of Analysis Wilson adds the phrase "back to back". Wilson does not define a "back to back" figure. His allemande is a bit rounder than a normal back to back but perhaps that is the closest equivalent.
This is an unusual meaning for "allemande", but, as best as I can understand, that is what Wilson describes.

According to The Regency Dance site in his book Campbell's 11th Book of New and Favorite Country Dances & Strathspey Reels there is a dance "The Bob of Fattercairn". But I don't have access to the original to verify that the figures match.

The Regency Dance site confirms that Campbell did publish a dance named "The Highland Plaid, or the Tartan Plaiddie" in his 3rd collection, but it does not describe the figure.

In Dale's Selection of the Most Favorite Country Dances and Reels he published a figure similar to the RSCDS's one:

Change sides and back again; Hands across and back again; Lead down the middle, up again; cast off and Allemand.

RSCDS Book 8 says this was "collected in Moray."

Wilson, on page 54, writes:

SINGLE FIGURE 1^st strain repeated 2^d. played straight thro' & D.C.
Swing with right hands round 2^d. Cu: then with left   lead down the middle up again & right & left with top Cu:

OR THUS Set & hands across with 2^d. Cu: set & back again   lead down the middle up again & set to the top Cu:

DOUBLE FIGURE (Tune played played twice thro as marked)
The 1^st. Cu: cast off at the same the 3^d. Cu: set & lead up then the 3^d. Cu: cast off & the 1^st. Cu: set & lead up to places   cross over 1 Cu: half figure round the 3^d. Cu: set contrary corners   lead thro the bottom hal pou^e.

Wilson usually gives multiple figures for each tune, the RSCDS appears to be following the DOUBLE FIGURE.

Wilson defines most of his figures in his An Analysis of Country Dancing.
Except that the first figure is not explicitly defined.
Cross over 1 Cu: is not defined either, but Cross over 2 Cu: may be found on page 22 and means "cross go below 2s, cross again, go below 3s". (so presumably crossing 1 Cu: means stopping below 2s.)
half figure may be found on page 54 and means "the active lady loops around the man below ending in her partner's place, as the active man loops the woman below ending in his partner's place. set contrary corners may be found on pages 55-57 and means "move to 1st corner and set, pass partner right moving to 2nd corner, set, return to places the easiest way". Wilson does not show the corners setting back.
lead thro the bottom is half of through bottom and top on page 23 and means "the active couple take hands, lead down below the bottom, drop hands, and cast up to their own places.
half poussette not defined. A full poussette is defined on page 53 going once and a half around the neighbor couple. But half of once and a half doesn't make sense here. Dancers have already progressed so going a full circle is probably what he wants to happen.

The RSCDS attributes this to William Campbell, and he did publish a dance named Tulloch Gorum, but I can't check whether the figure matches.

In 1796 Skillern published Tulloch Goram and later Skillern (again), and Bland and Wilson all published dances named Tulloch Gorum whose figures do not match.

Smyth and the Lowes (~1830) published a dance called Tullochgorum but again the figure does not match that of the RSCDS.

Allan (~1895) also published a Tullochgorum but his figure does not match that of the RSCDS.

RSCDS Book 9 says "collected in the Border Country."

Longman & Boroderip first published a dance called "The Duchess's Slipper" in 1790, and republished it in 1796 (with the same figure) in Twenty Four Country Dances for the Year 1796.

The 1^st. Lady hay with her partner and the 2^d. Gent: The 1^st. Gent: hay with his partner and the 2^d. Lady Lead down two Cou: up again one Cou: Allemande

The music consists of a four bar strain and an eight bar strain, both repeated.

As always, note the lead down and back is progress in the original, while the allemande is not.

The instructions say the 1s should lead down, but dancing down with nearer hands joined flows more easily from the two hand turn, and that's what the videos I've watched show, so that's what I'm doing here.

RSCDS Book 10 says "collected in Roxburghshire."

According to Hugh Thurston's Scotland's Dances, 1954 (on page 98) this dance was taught by Mr. William Seymour for Kilbride in 1805.

I do not have a description of the figure so I can't check that it matches the modern one.

In many ways this is a three couple dance in a four couple set with the 1s progressing one space down, but there are a number of whole set figures.

RSCDS Book 12 says "collected in the Border Country."

RSCDS Book 14 attributes this to Dale, in 1799, but I cannot find the original. However it was published by Cahusac in Cahusac's Annual Collection of Twelve favorite Country Dances With their Basses for the Year 1801 as The Devil among the Taylors., and he writes:

The 1^t. & 2^d. sett all four & right hands across do the same back again lead down the middle up again & allemand hands 6 round

The RSCDS attributes this dance to Skillern, but Bride published the figure first, then Skillern, Longman & Broderip, and Thompson published it as well. The only one I can find is Thompson, who writes:

The 1^st. Lady turns the 3^d. Gent 1^st. Gent. turns the 3^d. 1^st. Lady Allemand Right & Left with the 2^d Gent. at the same time the 1^st. Gent. Allemands with the 2^d Lady Sett 3 & 3 top & bottom the same sideways hands six round back again lead out sides

The original music consisted of 3 eight bar strains, all repeated.

RSCDS Book 15 says "from an old book of Scottish and Irish Dances (title page lost)."

Longman & Broderip published Platt's Six Cotillions and 12 Country Dances, for the Year 1791 which contains a dance with this name and essentially this figure (according to Robert Keller's site.

In 1792 Bland and Longman & Broderip published a different figure giving it the same name.

The crib diagram attributes this to the Blantyre MS which is dated ~1805. I cannot verify this claim.

In his A Companion to the Ball Room, 1816, Thomas Wilson published Lord Hume's Reel, wherein he writes:

SINGLE FIGURE (1^st. strain repeated 2^d. played straight thro')
The 1^st. lady leads down the 2^d. gent: the 1^st. gent: leads down the 2^d. lady   lead your partner down the middle up again & right & left with top Cu:  

OR THUS Set & half right & left with 2^d. Cu: set & back again   lead down the middle up again & set to the top Cu:  

DOUBLE FIGURE (Tune played twice thro' as marked)
Hands 3 round with 2^d. lady hands 3 round with 2^d. gent:   lead down the middle up again allemande   set 3 across set 3 in your places   lead thro the bottom & chain figure 4 round with top Cu:  

The music consists of a four bar strain and an eight bar strain. The RSCDS has picked the third figure (the one marked "DOUBLE FIGURE") which means the music should be played AABAAB. " " indicates two repetitions of a of the first strain, and " " indicates one repetition of the second (each accounting for 8 bars of music).

Wilson defines his figures in his An Analysis of Country Dancing. Many of them mean what you expect, but a few are surprising:

1. lead down the middle up again may be found on pages 10-11 and means: "lead down two couples, turn, lead up to the place of the couple who were below (who move up)" So this involves progression, and it only takes 4 bars. Wilson does not specify a hand hold for the lead, nor whether the 2s take hands.
2. allemande may be found on page 7 and appears to mean going about your partner in a circle. In a later edition of Analysis Wilson adds the phrase "back to back". Wilson does not define a "back to back" figure. His allemande is a bit rounder than a normal back to back but perhaps that is the closest equivalent. Note this is not progressive and only takes four bars.
   This is an unusual meaning for "allemande", but, as best as I can understand, that is what Wilson describes.
3. set 3 across & set 3 in your places is only described as "set three in their places and set three across" (opposite order) and is found on pages 76-77, and means: "Two lines of three across set, the ones move to their own places, everyone sets again." This takes 8 bars. Wilson does not say whether they take hands when setting. Wilson does not indicate whether the corners set when the 1s move.
4. Chain figure for four may be found on page 76 of the 3rd edition of the Analysis (1811) and boils down to what anyone else would call rights and lefts. This takes four bars.

Although my copy of RSCDS Book 17 gives no attribution for this dance, both the Scottish Country Dance Dictionary, and the Scottish Country Dance Database attribute it to William Campbell, but The Regency Dance site's list of his dances does NOT include one with anything like this name.

The RSCDS attributes this to William Campbell, and he did devise and publish a dance named Lady Auckland's Reel, but I can't check whether the figure matches.

Although my copy of RSCDS Book 18 gives no attribution for this dance, both the Scottish Country Dance Dictionary, and the Scottish Country Dance Database attribute this dance to William Campbell, 1795. According to the Regency Dance site's paper on Campbell's dances, Campbell never published a dance called Prince of Wales, but in 1795 he did publish (and indeed devised) a dance called The Princess of Wales's Strathspey, perhaps that is the origin of this dance. (I do not have access to Campbell's book to check the figure so I can't be sure.

This is one of the few books by Campbell which can actually be dated, he took it in to the Stationer's Hall in 1795 to register the copyright.

Wilson writes:

SINGLE FIGURE1^st. strain repeat 2^d. played straight thro & D.C.
Set & half right & left with 2^d.; set & back again   lead don the middle up again & half poussette with top Cu:

OR THUSHands 3 round on the ladies side hand 3 round on the gent: side   lead down the middle up again & set to the top Cu:

DOUBLE FIGURE Tune played twice thro' as marked
Swing with right hands round 2^d. Cu: then with left   set 3 across set 3 in your places whole figure contrary corners   chain figure four round with top Cu: & allemande

Wilson has a different figure for this tune in his earlier book Treasures of Terpsichore, 1809. Preston has yet another figure in 1796. Cahusac gives two figures in 1795. Campbell has a figure in Campbell's 9th Book of New and Favorite Country Dances & Strathspey Reels from about 1794. While Rutherford, 1756 and Johnson, 1742 give the same figure (Rutherford's compleat Collection of 200 of the most celebrated Country Dances both Old and New, Vol. 1) — but the music they give is a different tune (for one thing it is a jig, not a reel).

The following is an attempt to understand Wilson's DOUBLE FIGURE which is the basis of the Scottish Country Dance.

In An Analysis of Country Dancing Wilson defines some of the terms he uses. A "whole figure" is just a "full figure eight" (as you might expect, Playford uses the same phrase). Wilson does not explicitly define "whole figure contrary corners" in his first edition, but he does in his second on page 104 of the 2nd edition of the Analysis (1811). Rather awkwardly, Wilson provides two different (contradictory) directions. In the first "the 1s cross down, W1 in front of M1, W1 around M3, M1 around W3, then cross up, W1 around W2, M1 around M2, and return home". In the second the "1s cross up, W1 around M2, M1 around W2, then cross down W1 around W3, M1 around M3, and return home.

Chain figure for four is also defined in the second edition, on page 76 and it is what anyone else would call rights and lefts.

Wilson gives a very unclear definition of "allemande" it might be a modern gypsy, or a back to back (he defines neigther of these figures). He says the two people doing the allemande walk in circles around one another, he does not mention taking hands, he does not say how their faces are oriented.

Wilson, on page 96, writes:

SINGLE FIGURE Tune played straight thro
Cross over 1 Cu: half figure round the 3^d Cu:   & lead outsides  
OR THUSCast off 2 Cu: & back again   & whole poussette  
DOUBLE FIGURE Each strain repeated
Hey on your own sides   lead down the middle up again allemande   turn corners   & the double triangle  

This is an interpretation of the DOUBLE FIGURE. The music is two 8 bar strains (jigs) both repeated.

Wilson defines most of his figures in his An Analysis of Country Dancing.
Hey on your own sides is only defined as part of "Hey Contrary Sides, and Hey of your own Sides" on page 83 and he shows a mirror/reflection hey.
lead down the middle up again may be found on pages 10-11 and means: "lead down two couples, turn, lead up to the place of the couple who were below (who move up)" Wilson does not specify a hand hold for the lead, nor whether the 2s take hands.
allemande may be found on page 7 and appears to mean going about your partner in a circle. In a later edition of Analysis Wilson adds the phrase "back to back". Wilson does not define a "back to back" figure. His allemande is a bit rounder than a normal back to back but perhaps that is the closest equivalent.
This is an unusual meaning for "allemande", but, as best as I can understand, that is what Wilson describes.
turn corners may be found on pages 28-32 and appears to mean: the 1s pass partner right shoulder to turn first corners with both hands, then pass each other right shoulder to turn second corners (presumably again with both hands, though Wilson does not specify), finally passing each other left to return to places.
the double triangle does not appear until the 3rd edition of the Analysis on page 113 and means: the middle couple loop first corner right shoulder pass outside partner's place, then loop 2nd corner right shoulder and return to place.
This is completely different from the RSCDS version of double triangles.

According to the the Regency Dance site Wilson devised, and Button & Whitaker published, a dance with this name but whether its figure matches the RSCDS's, I cannot say.

Since the dance contains double triangles, it seems likely Wilson was its deviser.

The RSCDS attributes this dance to William Campbell, and he did publish it, but he attributes it to "D. Mackenzie".

On page 17 of book 10, Campbell writes:

The 1^st. & 2^d. Cu Reel on the Ladys side the same on the Gents. side the 1^st. Cu. lead down the middle up again to the top and Allemande.

The music consists of a four bar strain and an eight bar strain. I assume it is intended to be played AABB.

I think the RSCDS's interpretations of the reels is unlikely, given how crowded that would make the duple minor set.

According to the the Regency Dance site Campbell did publish a dance with this name. Whether the figure matches I cannot check.

Campbell writes:

1^st. Cu set & change sides Back again — Lead down the middle up again to the top then 1^st. & 2^d. Cu hands across half round back again — Poussette quite round with the 2^d. Cu hands 6 round Allemande.

Where does Campbell mean the progression to happen?

set & change sides Back again is not progressive.

What about Lead down the middle up again to the top? The "up again to top" does sound as if it returns the 1s to their original places. Yet if you look at Campbell's The new German Waltze (also in book 10) you will see him use those same words where that is the only move which could be progressive.

So I conclude that neither Poussette quite round nor Allemande is progressive.

Campbell's hands 6 round suggests a 3 couple dance, but all the other figures are either for the 1s+2s or the 1s alone, Changing the circle six to circle four easily makes for a 2 couple dance.

The Scottish Country Dance Dictionary attributes this to "J Cameron".

In 1821, GMS Chivers published a dance with this name but a different figure. Boulogne (1827) Smyth (1830), and the Lowes (1831) did the same. Elias Howe, in Boston, 1858, published a dance with this name and a figure with some similarities, probably not a source but an example of a dance which evolved from the same root.

John Bowie calls this dance Miss Murray   Lintrose, writes:

Cast back one Couple Almand half round; cast back another Couple Almand half round; Lead up the middle cast of: Lead down one Couple & cast up one Couple: first Lady Almand with 2^d. Gent: first Gent. Almand with 2^d. Lady:

Goulding spells the title The Land of Cakes.

The adaptation is rather loose.

Straight writes:

Change sides & back again lead down the middle up again & cast off allemand with the right & left hand hands 6 quite round

SINGLE FIGURE Tune played straight thro
Set & change sides with 2^d. Cu: set & back again   lead down the middle up again & right & left with the top Cu:  

OR THUS Cast off 2 Cu: & back again   & whole poussette  

DOUBLE FIGURE Each strain repeated
The 3 ladies lead round the 3 gent:   the 3 gent: lead round the 3 ladies   set & change places with 2^d Cu: allemande   & whole figure contrary corners  

SINGLE FIGURE Tune played straight thro
Hands across with 2^d. Cu: & back again   lead down the middle up again & allemande  
DOUBLE FIGURE
Hey contrary sides   hey on your own sides   swing with right hands round the 2^d. Cu: then with left   chain figure 4 round with bottom Cu: & half poussette with top Cu:  

Wilson spells the title as "Captain Macintosh".

The music consists of two 8 bar strains.

The RSCDS is interpretting Wilson's first DOUBLE FIGURE

Wilson defines most of his figures in his An Analysis of Country Dancing, 3rd Edition.

• The 3 ladies lead round the 3 gent: may be found on page 16 and means: "all the ladies take hands, lady 1 leads counter-clockwise above, and then behind the men, at bottom looping back to original places."
• The 3 gent: lead round the 3 ladies may also be found on page 16 and means: "all the men take hands, man 1 leads clockwise above, and then behind the ladies, at bottom looping back to original places."
• set & change places is defined on page 51 and means: "the active couple face the couple below, a set to them (they set back), then change places passing right shoulders. (Wilson also shows how to change back)
allemande may be found on page 3 mean going about your partner in a circle, "back to back". Wilson does not define a "back to back" figure. His allemande is a bit rounder than a normal back to back but perhaps that is the closest equivalent.
This is an unusual meaning for "allemande", but, as best as I can understand, that is what Wilson describes.

• whole figure contrary corners is defined on page 104. Rather awkwardly, Wilson provides two different (contradictory) directions. In the first "the 1s cross down, W1 in front of M1, W1 around M3, M1 around W3, then cross up, W1 around W2, M1 around M2, and return home". In the second the "1s cross up, W1 around M2, M1 around W2, then cross down W1 around W3, M1 around M3, and return home.