This question would depend on the mentality of the person involved, of course. I imagine that most introverts would rather be rejected for who they are, while extroverts may feel more comfortable being accepted for who they aren't. It depends on how deeply we delve into things.
Being accepted for something we're not may act like a temporary fix, but it'll eventually take its toll. Last year I discovered that one of my close friends was gay...after he attempted to commit suicide. He always seemed happy, and everyone thought he was a well-adjusted, straight young man. He was accepted for who everyone thought he was, but he couldn't accept himself.