I wasn't referring to you and me; I was referring to him and me. We disagreed on several topics, and I'm sure each of us wrote some things that would be called heated at one point or another.
No. I don't know what else you could call a long and drawn out debate like this one. Debate is pretty much another word for argument.
I'm not "trying" to be overly correct. I seriously cannot see any other answer. If you have one, I'd be surprised to hear it.