I'll react differently if they're insulting me, vs. someone I care about. I think it's easier for me to back down when they're insulting me. They'll make some accusation about me, I'll immediately get defensive, and if they don't back down (i.e. if they absolutely refuse to listen to reason), I'll assume that they can't be reasoned with and I'll give up -- and if I'm especially angry, I'll make sure I have a good zinger to use as the last word before I storm out of the room.
But if they insult my friends, I will argue with them for as long as it takes for them to either change their minds or back the fuck off.
Never, ever, will I raise my voice with them. My arguing will be very testy and with gritted teeth; at most, I'll seem very frustrated, but never full-out raging. And I will never try to fight with them; I will try to change their minds. When I fight, my definition of "victory" isn't just that they're defeated (although that's nice too) -- my definition of "victory" involves the enemy changing sides.