I really like this. A few minor suggestions, with notes:

“…The dark side is your birthright. Rey…”
“You’re lying.”
“You were right about your parents, Rey. They were nobody. And nobody conceived you.”

This rewrite puts Kylo’s information further into the perspective of a continuation of the lines about her birthright. And I think it’s a little more elegant, if cryptic (though that’ll become clear in this conversation).

“Luke saw the same fate in me. But together we can resist that fate.”
“Stop talking.”
“I’m the only one who truly knows you.”

Slightly more elegant again, and maintains Kylo’s characterisation that he feels betrayed by Luke, and leads into his later “join me” line.

The only other thing I’m not quite sure about is Leia. I think we want to keep her in as high esteem as possible, even though she did reject the path. She shouldn’t doubt her own ability to resist the dark too much- but another of her positive character traits has been her pragmatism.

“She was quick to learn in our training. But Leia knew that she could do so much good without needing to tempt the dark within. She surrendered her saber to me and said that one day it would be picked up again by someone with the spirit to face that darkness.”

Not quite sure about the wording here, but open to iteration on this idea.