This dance (Single Circle) has no relationship to the dance "Circassian Circle" in RSCDS Book 1 #3.  (Couple facing Couple ).  Many variants of this dance exist, but the original source is unknown, and might not exist. 

This dance has been entered using the video of the Edinburgh team's display in Newcastle 2012 as source.  The Edinburgh Branch seem to have worked from online sources, and say that they used a “variation” of this dance. 

A broad overview of dances called "Circassian Circle" can be found on Andrew Timmins' website at https://www.libraryofdance.org/dances/circassian-circle/

Supplement:  The website https://www.dancilla.com/wiki/index.php/Circassian_Circle  tells us that the Community Dances Manual 1, 1949, (still available from the English Folk Dance And Song Society; Cecil Sharp House, 2 Regent's Park Rd, London NW1 7AY.) has the following entry where both dances are combined:

13. The Circassian And The Big Circle (Collected by Maud Karpeles in Northumberland)

PART 1: The Cercassian Circle (jig tune) Form: Progressive Sicilian Circle (couple facing couple). A.1. Right and left through or woman cross over (left) and men cross over (right), and repeat to places. A.2. Partners balance right and left. Partners swing. B.1. "Ladies Chain" B.2. Swing and change OR Promenade on to next couple.

PART 2: The Big Cercle (reel tune) All join hands in one ring: Each man with his partner on his right facing centre. A.1.Forward and back twice. A.2. Woman to centre and back again. Men go in and back to contrary partner, i.e. the woman on his left. B.1. Swing her round. B.2. Promenade around and repeat as from A.1." 

Could this double variant be the reason why these unsimilar dances have such similar names?