As a trans nonbinary person, yeah, my definition of trans is basically "anyone who is not cis who wants to identify as trans." There are a variety of reasons someone might not identify as trans, some more valid (there's probably a better word, but) than others, and it's ultimately up to the individual.
When I first started identifying as nonbinary, I didn't identify as trans because I felt like my experiences weren't "enough" and that I would be coopting others' struggles if I claimed trans identity. I also tried to explain it in a semantics-argument type way that focused more on how words were formed than how they were used. Anyway, my reasons were bullshit. In part because of that, I do encourage anyone who's new to identifying as not cis but not trans (and interested in exploring) to think deeply about why they don't identify as trans. And if they come out of that still not feeling trans as a label, cool. And if that changes down the road, or changes back and forth a few times, also cool.
When I use "trans," I also automatically include trans nonbinary people, like how when I say "women," I include trans women and cis women. It bothers me when I see people assume "trans" means binary trans people or use it that way. That's usually cis people I see that from, even in spaces that a generally pretty understanding. Like, it's happened here (not this thread here, this forum in general), in the DoA comment section, etc. There, the person I was talking to said there should be a way to distinguish whether you're referring to binary trans people or nonbinary people, and, it's like, there is! Say "binary trans people" if you're trying to make that distinction. But often it's not as necessary or helpful as some people might think.
I appreciate the phrase "trans and/or nonbinary people" for seeking to be inclusive (though I still know people who identify outside of that), and with the "and/or," I think it's good. I think some people come away from it with the impression that nonbinary is inherently a separate thing, but there's only so much you can do to curb misunderstanding. I don't like when I see "trans or nonbinary" because while I know that the "inclusive or" can exist, I think people default to assuming the "exclusive or." (God, are there other terms for that? It's been so long since I studied that stuff.)