I think some of the replies have gone away from what the original question was (which I highlighted in red italics above). The question wasn't "which one is better", or "which one is right". It was how each of us defines role-playing, and he added "in character" as well. IMHO, one of the most important aspects of role-playing is being able to divorce yourself from "player" knowledge and sticking to "character" knowledge. I have made numerous decisions that "roll" players and munchkins would have chewed me out for, saying that they were stupid or tactically poor, but I made them using the knowledge that I knew my character had, even though I had more knowledge that would normally have altered my decision.
Roleplaying to me is getting to know who your character is, what his motivations are, and immersing yourself in the character and situations.
Depending upon the game system being played, sometimes there's more dice involved, sometimes less, but being able to play the ROLE, I think, is what truely separates the roleplayer from the rollplayer.