I think that becoming an achievement/trophy whore entails playing a game only for said achievements/trophies, and not having fun--turning a game into work, if you will. I can't say I've never gone that far, though. I remember going 99 laps on a racebox track in Wipeout HD just to have the trophy for it, but it was a chore and I didn't enjoy myself very much.
However, while I do still strive to eventually get the trophy for maxing out a class level in Resistance 2 online co-op play, I get to enjoy myself while playing, regardless of how much experience I get. The trophy is merely a secondary prize when compared to the equipment upgrades for the class that help make the playing experience better. I understand that trophies and achievements are there to encourage players to explore and appreciate more of the game that the developers put so much effort into making, and sometimes it helps, as in LittleBigPlanet when you're rewarded for participating in the online community for things such as commenting on a user-made level or playing with a full party of players, but again the game is enjoyable and rewarding on its own.
So I suppose you're only really an achievement/trophy whore if you're playing a game solely for those rewards and not having any fun with the game itself.