Yeah, when it's check it's the Moscow, and when it's not check it's the Rossolimo (not check because a knight on c6).
Canal is probably just another name for the Moscow.
And of course you can transpose from one to the other.
Rossolimo is more main stream. As white you often give black some structural problems. It was played a few times in the 2018 WCC match.
Moscow's (canal's) reputation is it's solid, but unambitious for white.
I recently played two Sicilians. I thought the Rossolimo was the Canal Sokolsky. I have also seen it referred to as the Moscow variation. Are there differences or is it just inconsistent labeling?
This site has a video which seems to put them all under one label. https://store.chessclub.com/videos/icc-opening-videos/har-zvi-sicilian-moscow-variation B51, B52: Sicilian: Canal-Sokolsky (Nimzovich-Rossolimo, Moscow) attack.
So, can someone explain why this labeling is shown in two of my games?