In other words, people who moved on from the gender assigned to them at birth.

Some non-binary people identify as transgender but not all of them do. Just like non-binary is both an umbrella term and a spicific gender identity, transgender is both an umbrella term and a specific gender identity. You'll often see the shorthand "trans*" to describe the umbrella term and the shorthand "enby" to describe the specific gender identity.

Examples of clear gender identities that are not used as umbrella terms are: trans women, trans men, genderfluid, agender, demigirl, etc.

Yes, and the Latin word trānseō (nominal form trānsitiō → English: transition) means exactly to "go on across", "go beyond". Trāns ("across, beyond") + eō, it ("I go, he/she/it goes") + tiō (→ English: "-tion"). It's a distinction without a difference.

What does it mean to "identify" with a claim about what society is like - namely that "binary" gender might be more of a spectrum, with weird liminal stages in-between? You've said that this is what "non-binary" is about. How does this even begin to square with all those other notions about gender identity being something exceedingly clearcut, that someone can base major life decisions on?