There is no "right answer" to the following, and generally I don't think as much about this as others do.
I don't like many of the labels people with ASDs have adopted. I have no idea why, but I don't care for autie, aspie, aspergian, or several other terms I've seen online and in print.
But, I don't also don't know if it is better to write: "person with autism" or "autistic person" because to me these are like comparing "person with blue eyes" and "blue-eyed person." I suppose a better comparison is "person with deafness" and "deaf person."
My traits are "autistic," but several people have responded negatively to "autistic" as an adjective. I don't know that "autism" is a great noun, though, since we have no idea what that means. Autism itself is defined by traits -- it is necessarily an adjectival label.
I believe there are likely many "autisms" with various causes. The similar traits are "autistic" traits. That's why I'm okay with "The Autistic Me" compared to "The Me with Autism." That implies there are times when I'm not with autism -- maybe I'm with hyperactivity. (I certainly could be called "The Hyperactive Me.")
As I said, I don't have any answers, only personal preferences.