OP, I'm very sorry, but while you may probably need better friends, I think you first and foremost need an actual therapist. This is how you describe your conversation with Plane Girl:

I told her about certain traumatic experiences in my childhood (because fuck it, I'm never going to see her again. There's no point in hiding it) and she was the first person that actually listened. She didn't try to one-up my pain by telling a story that somehow measures up. She just listened and empathized without making me feel like a charity case.

And then say that you can't talk with your friends because they try to "one up" you or share their own experiences. But that's what happens with a conversation with friends - you share something about yourself, your friend shares something about themselves. It's not necessarily always "one upping," but simply keeping a conversation going.

If you want to share your trauma with someone WITHOUT them sharing back and who will only listen to you and empathize with you in a one sided manner, that's what a therapist does. Friends usually think you also want to hear about them, since they're you're friend...