My daughter was exactly the same at age 2 and has completely flip-flopped at daycare.
She was initially very reserved around kids (and still is around new adults), but is very social with the other kids in her daycare classes. It took a couple of years, but her personality and openness to other children is now commented on by her teachers and other adults.
She actively attaches herself to new kids in class and integrates them into the class. She's still not a very aggressive socializer with new kids in new environments, but she is much more open than I would have ever imagined.
Good luck - I don't have any advice other than to continue to trust your instincts, but wanted to let you the "fitting in" thing might change in unexpected ways.
She was initially very reserved around kids (and still is around new adults), but is very social with the other kids in her daycare classes. It took a couple of years, but her personality and openness to other children is now commented on by her teachers and other adults.
She actively attaches herself to new kids in class and integrates them into the class. She's still not a very aggressive socializer with new kids in new environments, but she is much more open than I would have ever imagined.
Good luck - I don't have any advice other than to continue to trust your instincts, but wanted to let you the "fitting in" thing might change in unexpected ways.