HowlerKarma, you are describing her extremely well:)
You're right, her behavior is not a concern to her. I've told her that she's missing out if she behaves like that and she just laughs at me in a "yeah right" kind of way. She says she's very happy being by herself at times. And yes, she does cut loose with people she knows well and at home. It's hilarious how different she is.

I find Dudes observation very interesting. From the time she was just a few weeks old my daughter would "lose it" if a stranger would pick her up. Once she was a few months old, she didn't even like strangers looking at her from a close range. My younger child was the total opposite. Never cared who picked him up or held him or talked to him. And he can still approach and strike up a conversation with anybody, anywhere, anytime.

My DD is just too serious most of the time. She absolutely despises singing and dancing, she thinks it's embarrassing and is refusing to do it. Any kind of holiday concerts or classroom dance/singing activities where participating is not optional, is downright embarrassing for me to watch, because she sticks out like sore thumb by just standing there. And the teachers might take it as if she's stubborn, when in reality she just can't make herself do it because for her it's debilitatingly embarrassing. Now she can step out on a stage in front of hundreds of people and perform on one of her instruments without even flinching or perform a theatrical skit in front of the classroom with no problem. It does not make sense to me.
So should I just let her be? Is she missing out without knowing it herself? Is this behavior holding her back and reaching her potential? Is it going to get worse if I don't intervene?