I think that a core problem is that upwards of 90% of players in this hobby want their characters to be Cool. There are players who enjoy playing weird little gremlins or satirical characters they see as totally separate from themselves, where they get to have fun at that character’s expense. These tend to be my favourite players, but they are in the minority. Mostly people have fantasies about being Sexy or Strong or Smart or Cool that these characters, and their wins, are meant to fulfil.
So as much as we can look down on people who see their character’s loss as their loss, for having trouble separating IC and OOC, the reality is that this is most people. And even if you are someone who enjoys playing the foil or the weird little gremlin, I’ll wager that at some point you’ve realised that what most people enjoy about your characters is they get to feel Cool in comparison, when they’re putting your character down or beating them up or being morally upstanding in contrast to your character. And you’re OOCly giving them the thumbs up that no feelings are being hurt in the process. That’s why playing what most people consider a “good villain” is basically a service you’re providing other players, because they get to feel Cool rooting for their own character against yours, while you’re always holding back just enough that it never feels humiliating or futile. This is CvC, but you’re serving the needs of P through a C veil, and if you stop doing that, the feelings that arise are PvP.
So staff can call it CvC, but they need to understand that it’s still ultimately a balancing act of managing player egos.