Source: Traditional, collected by Maid Karpeles, published by Peter Kennedy in English Dance & Song, February 1948. Vol XII. No. 1
||Formation: Sicilian Circle
Right & Left Through OR Ladies Cross by the Left Shoulder; Men Cross by the Right Shoulder
Repeat to places
||Partner Balance & Swing
||Ladies' Chain x2
||Swing & Change OR Promenade on to the next couple
||Formation: Circle Mixer
||Into the Middle & Back x 2
Ladies into the Middle, Clap, & Back
Men into the Middle, Clap, & Turn Left and go to your Corner
||Promenade - this is your New Partner
As published in the Community Dances Manual 1 in 1949: "A jig for one part and a reel for the other." It is generally danced these days to whatever the band want to play.
Part 1 has disappeared almost completely from the modern dance floor, but it was the original Circassian Circle.
Part 2, however, is a very common dance, often used as a finisher, with the dancers being told to find their original partner during the last B1, ready for a final swing. Part 2 is widely known as just "Circassian Circle". It was collected by Maud Karpeles in 1928.
For Part 1, the earliest publication that I know of is by Coulon in 1844; this describes the formation, couple facing couple in a big circle, but all it says about the dance is, "The figures to this dance may be taken either from the country dance, or from the quadrille: the waltz figures may also be introduced with advantage."
Radestock, in 1877, specifies that the first figure of the Caledonian Quadrille is generally danced:
Top & bottom couples hands across
Set to your partners and turn
Half right and left
Later 19th century sources give descriptions for Part 1 such as:
All balance (4); Four hands around (4)
Ladies' chain (8)
All balance (4); Turn partners (4)
Right and left (8)
All forward and back (4); Forward and pass through to next couple (4)
There are lot of common elements between these early forms of Part 1, and the one that Maud Karpeles collected, with various moves being modernised, e.g. Turn becomes Swing. The original Ladies' Chains would have been with an Allemande Left rather than a Courtesy Turn.
Basically it would appear that, originally, "Circassian Circle" just referred to the formation, and therefore had the same meaning as "Sicilian Circle".
For further reading see:
Library of Dance
Maud Karpeles notes