What Jojo did to Sylvia first WAS Carnymancy. We don't know how he eventually saved her, but we know that keeping her alive for multiple turns while Incapacitated was Carnymancy. So that's one to add to the list.
The phrase/theme of "rigging the game" is what I feel inspires the broad definition of what we assume Carnymancy does, combined with the name itself. Schools of magic include literal and satirical interpretations of their name. Carnies in stupidworld cheat. Carnies in erfworld admit it's the magic of rigging the game. To me that points to manipulating with existing mechanics as being their primary power.
"I'm afraid I don't understand. And also afraid that I do."
Two guys with basically the same name in a discussion about a character getting cloned.
There's gotta be a good joke in here somewhere.