That's pretty much it. Years ago, Kakashi told team 7 that everybody dear to him is gone (I think it was said to semi-emo Sasuke before he left, and why he taught him the Chidori).
He considers Gai his rival, but certainly not is best friend.
You probably don't have to directly kill your friend, but I think you have to feel responsible for it in return. Otherwise, how many Uchihas have had their best friend killed?
Gai sees Kakashi as a rival, Kakashi couldn't care less about that.