Battles Won

Are you just showing off your 6000 battled Vaporeon!? I don't think there's another way to see it but I'm impressed :).

While your Pokemon is defending a gym, if you check your badges, it'll show a profile of your Pokemon with it's remaining CP/motivation and how many battles it's won.

It looks like when you're going off into battle the number they show is how many times you've used that Pokémon to defeat another. As for battles won defensively, I have no idea. I'm wondering as well