I'm very much like you in that many people mistake me for an extrovert because of the opportune occasions where I display "extroverted" traits in social settings. However it's more of a decision (mostly based on energy) than a default mode I fall back on.
I'm not sure this has anything to do with introversion.
Speech and debate clubs are filled with introverts. The notion of "stage fright" is completely orthogonal to introversion. You're speaking at people, with YOUR ideas, YOUR words - it's not a dialogue, with shared feelings, mood, etc, which is where introverts generally get drained.
The whole discussion here is about introverts and how the existing public perception of us is flawed, because it misses the point entirely. As playing musician I know many extrovert performers who are agued with stage fright, something which doesn't bother me as an introvert at all.
Maybe it is because being seen in a bad light by people I wouldn't enjoy to meeting in the first place isn't a threat. When I play, my biggest fear is that we won't perform to our/my own standards and thats it. If the audience digs it that is great, but I am not afraid of them hating it, because I am aware of how good/bad what we do is.
What makes me an introvert isn't that I am shy (I am not), what makes me an introvert is that I find all interaction with other people except a select few exhausting and draining. Don't get me wrong: That doesn't mean I am weird or bad at talking with people — in fact I am qite the opposite. It just means it costs me a lot of energy, while extroverts appear to lose energy when they are alone, I lose it when there is a certain kind of social interaction needed around me.
So people don't really see me as an introvert, because to them it is all about being shy and a little bit awkward. But that isn't what makes you an introvert, the energy part is.