> The reason it hurts her and she gets frustrated and angry about my lack of progress is because she does care about me.

Sounds like... Your therapist judges you 'you haven't made any progress' and then she blames you for feeling frustrated and angry.

Instead of just... Accepting you.

> I feel like I am so insignificant. I will never belong anywhere, and I will never have the family that I needed in childhood.

And her failure to accept you makes things worse. Because if she won't accept you and delight in your strengths just as you are then what hope is there that anybody else will?

> How can someone who isn't going to be a permanent part of my life fill those needs?

By accepting you.

> I don't want to mess up anymore. I want to get it right, and I want her approval, but trying to get it right has just gotten me in trouble before. I don't want to be in trouble anymore.

I've thought for a while now... That it doesn't sound like your therapist is very accepting of you. Of course, you aren't very accepting of yourself, but part of her job is to show you how to accept yourself. But that isn't going to happen if she doesn't know how to accept you either. I figure that you were probably drawn to her because you feel like you need a 'good talking to' and your therapist thought that giving you a 'good talking to' would help you.

But really... It is just an old old cycle of people not accepting you and demanding that you change and threatening to withdraw their love from you if you don't change. Familiar ground for you... But the answer isn't your trusting and accepting that that third part won't happen. The answer is your learning that people can accept you right now just as you are. And that... You can learn how to do that too.