how to obey battles won?

When the last Pokemon of an attacker's lineup faints to that defender, I believe.

Leaving early doesn't increase the count.

Androgynous is correct as far as I know but it also seems to be glitchy. I've put a freshly evolved mon in a gym (100 percent sure it was never used before) and it already had "1 battle won" I might add I used the mentioned Pokemon to clear that same gym.

Also timing out will result in adding to the count.

It only counts to the top defender at those timings, no matter which Pokémon does the attacker lose to.

Theoretically what androgynous said. However, it has been tested many many times by many players and the count hasn't been able to be increased under testing circumstances.

This would indicate that either it is really buggy and inconsistent, or that there are some hidden requirements to raise the count that we don't know of yet.