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.

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.

In his A Companion to the Ball Room, 1816, Thomas Wilson published Lady Mary Douglas:, wherein he writes:

SINGLE FIGURE Tune played straight thro and Da Capo
Whole figure at top lead down the middle up again turn your partner & lead outsides

OR THUS Cross over 2 Cu: cross up 1 Cu: towards the top and half figure round the top Cu: & the double triangle

DOUBLE FIGURE Tune played twice thro and Da Capo
Hey on your own sides chain figure 6 round lead down the middle up again lead thro the top Cu: whole figure round the bottom Cu: turn corners whole figure contrary corners

The music consists of 2 eight bar strains. The RSCDS has chosen the DOUBLE FIGURE which means the music should be played ABBABB (I think). "" indicates a strain.

Wilson defines 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.
• Chain figure of six may be found on page 78 of the 3rd edition of the Analysis (1811). The top couple face their partner, as the others face their neighbor then everyone does rights and lefts until they return to where they started. Wilson does not specify how to fit the 6 changes into 8 bars of music.
• 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.
• lead thro the top Cu: is half of through bottom and top on page 23 and means "the active couple take hands, lead above the top, drop hands, and cast up to their own places.
• whole figure round the bottom Cu: is defined as whole figure at top (that is when the 1s are at top so they figure through the couple below them) may be found on pages 12-13 and means "the active lady starts and crosses down, looping around the man below her then she again cross between the couple below, looping around the woman below and casts up to her own place. The active man starts a little after her and mirrors her motions, looping the woman below and then the man below."
• 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.
• whole figure contrary corners is found on page 104 of the 3rd 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.

The RSCDS attributes this dance to Johnson (who published it in ~1748), but Walsh published it first.

Walsh writes:

Hey on the Men's side Then foot it to Partners then to sides turn single after Each then the 1st Man turn 3d. Cu. half round and cast up into the 2d Place The Wo. do the same back again change sides and foot it cast up change sides and foot it cast off

The music consists of two 4 bar strains.

Johnson published this in his 5th volume of Country Dances (1750), and Rutherford published essentially the same figure in 1759. Both Johnson and Rutherford spell this dance as "Willie with his tartan Trues". The RSCDS credits this dance to Rutherford

Hey Contrary sides Hey on your Own sides Cross over two Couple Lead up to Top, foot it And Cast off Foot it Corners & turn Foot it the Other Corners & turn Lead out Each side & turn your Partner

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.