I don’t know that the term CvC solves any problems, but I also don’t see it as spin. In my mind, calling IC conflict where everyone is working together to tell a good story CvC reframes the conflict to one where the players can be cooperative.
PvP doesn’t suggest to me that the players are going to start hitting each other. To me, the difference is that in PvP, the players are trying to one-up one another OOCly as well as ICly. They aren’t cooperating to tell a story, they’re pitting their OOC wits against one another as well as their IC wits (there could even be some situations where there’s OOC competition even though there’s IC cooperation, maybe two characters are working together one a project and are ICly cooperating, but the two players each want to one-up the other and come out looking “best” from the cooperation).
And PvP can be fun as hell, don’t get me wrong. There is indeed a different rush to knowing that you got one over on another real person, not just that your character won the dice rolls.
I don’t think that CvC should suggest that something is more or less messy than PvP, I think that it should suggest that the players are cooperating even though their characters are competing.