My DD8 is definitely an extrovert, as she's energized by being around other people and being the center of attention... but she has exhibited all the same behaviors indicated in the OP, of needing time and space to acclimate to new environments, activities, and especially, people. This was obvious even as an infant, when we had to insist that nobody pick her up until she indicated she was ready for them.
When meeting new kids in a park-like situation, she's been slow and reluctant to warm to some, and instantly fell in love with others (with the attendant heartbreak when it was time to leave). In the former situations, what helped her ease through them was the fact that she was playing with me, so that gave her something of a security blanket while she continued to ponder the new social situation. If one of the other kids tried to talk to her and she ignored them, I'd gently prompt her to respond, and I'd also clue her into some social cues she might have missed ("Hey, that kid wants to play with you"), but otherwise, she'd get no pressure from me, and I'd just let the situation develop organically.
Sometimes the other kids were more interested in playing with me than with DD.
