This dance (Couple facing Couple ) has no relationship to the dance “Circassian (Big) Circle” (Single Circle) . The original source of this dance is unknown, and might not exist.
Many variants of “Round the Room” dances called something like “Circassian Circle” exist, both with a single circle of dancers, and two circles (Couple facing Couple). A broad overview 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:
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?