While I don’t disagree with your take on “Just Rey,” I don’t agree with how you feel about “Rey Skywalker.” The beginning of the film isn’t just her being ashamed of who she is, she’s thinking she’s inadequate. She doesn’t deserve the saber. To me, her taking the Skywalker name is her saying “I am good enough to be a Skywalker, I’m not just Rey, I’m more than that.” It’s a similar sort of growth I think.
That’s fair. I think it’s what the movie was going for but it didn’t land for me. I definitely felt something was off in the theater but my thoughts on this weren’t really crystallized until I watched this Just Write video: https://www.youtube.com/watch?v=BD5mLw0A8vI&t=7m25s
The whole thing is great but I just linked to the part I feel is relevant here.