Read this thread before starting the event, but realized how the rare counter works a bit late.
So basically this is the what you should see to get the kill to count:
Which only happens if it dies by your killing blow or someone else's during your attack turn, as mentioned by Zack666.
It really does suck when you've just started and realize this too late. I've played IWZ, this game is basically the same but with a dreadful UI and a bit deeper progression from what I can tell. I could have been close to done with it if I went in knowing this but I'm currently at lvl 15 strength for finding rares and only 11 out 20 for the count. Managed to get 5 of those through joining a reinforcement call and either getting lucky with the killing blow/turn or timing my attack. As for my own rares, I have to burn through like 7+ Full BP recovery potions to kill one on my own.. it's really crappy they did it this way. ~(>_<~)