Pokemon-go – What does “Battles Won” refer to

This designated how many battles this Pokémon has won. In other words how many times it has successfully defended an attacker.

For further clarification, based on personal experience, I have only seen it pop up for Pokémon that took down my last Pokémon in my party when I was attacking a gym.

A few sources that back this:

Pokémon also display a "Battles Won: X" number that shows how many times that specific Pokémon has stopped an attacker -source

For each Battle Won, it means that your Pokemon was responsible for them losing the fight. That means their final attacking Pokemon was defeated by your guy. -source

Update: a change to gyms may have changed how this works. According to this reddit post,

I'm not sure if this is widely known, but the "Battles Won" that shows up on the pre-battle splash screen is not how many times that attacker was the victorious Pokemon when you defeated another Pokemon. It is actually how many times that attacker was part of a team that defeated a defending Pokemon. I noticed this because a Machamp I just caught from a raid had 15 victories this morning and I was pretty sure I hadn't used it that many times already. The 15 victories, though, referred to the 5 Pokemon I had beaten in a gym 3 times each last night even though I didn't use Machamp for each individual fight.

however, this is the only source I can find noting this change, and have not been able to confirm this myself.