The argument is that you shouldn't base your position of acceptance on whatever the DSM says because they're demonstrably very fallible, but rather you should use your own arguments instead.
The DSM is fallible. Many of the diagnostic criteria will change in future versions of the DSM. In the future we are going to learn that some aspects of the current manuals are wrong, just as we have for previous ones.
But the people who will find those errors and make those updates will be, and have been, researchers and mental health experts who work in the field and have relevant experience. There's no way that I, as some rando on the internet will have anything insightful to say on the subject. If either of us come to a conclusion that contradicts the DSM, it's far more likely to be we are mistaken in our uninformed opinions.
Regardless, the DSM-5-tr is the manual that is CURRENTLY used by mental health professionals to diagnose mental disorders. My description of the DSM criteria is as accurate as is practical for a single sentence sarcastic comment in response to a meme.
It is possible to accept people, use preferred pronouns and names etc. while also being of the opinion of "technically you're not trans but enby" or something. Some people just have a fetish for precisely defined taxonomies, don't kink-shame.
I'm sorry how is clearly delineating a clade (in this case, "trans") not crisp taxonomy.
You might, for example, come across a random dog and say "That's not a Rotweiler that's a dog": It might be another named breed, it might be an incomprehensibly mixed-up street pupper, point is it's not a Rotweiler but still a dog.
Ok, but even if we follow your dehumanising analogy, non-binary people are still trans
So claiming "“technically you’re not trans but enby” is never going to be "precisely defined taxonomy", no matter how much you'd love for transphobes to have their "kink" of deliberately mislabelling people to exclude them from a category they factually belong to.
Transgender = a person who identifies as a different gender to what they were assigned at birth. This includes non binary people, as well as agender, gender non conforming, and so on.
You wanting to exclude yourself from the trans community for whatever reasons (almost certainly a combination of internalised and/or regular old transphobia) doesn't change that.
But hey, don't take my word for it, you could have easily found that out for yourself if you weren't so intent on being "not like other trans people":
Oh wait sorry I assumed the other inclusion direction, that trans folks are enbies -- it's not what you said, but how I would organise the taxonomy if I was the taxonomy Emperor (which I am not). Though just having three categories (cis, trans, enby) without inclusion relation also makes sense.
category they factually belong to
All categories are man-made. Philosophers and Linguists would have a field day with that one.
Let's deescalate a bit. Anecdotal case, please don't ask me for the source, but once upon a time there was someone identifying as a trans man. Got top surgery. After some soul-searching and being thoroughly dissatisfied with the bottom surgery options available, he decided "aw shucks" and decided to henceforth identify as a butch lesbian, saying "The issue wasn't so much having a pussy but having a dick that's not mine in vicinity to it". Goes by she/they.
Cis, enby or trans? Gay or straight? At which point in time? Snowflake? (well, who isn't).
Stuff like that is why it's important to remember to not make categories normative -- both in the "you are X therefore you must do Y" sense, as well as "If it is bipedal and has no feathers then it's a human" kind of sense (which includes plucked chickens, ask Plato). They are descriptive at best, and noone uses the exact same definitions. Some say that's hot tea, other say it's warm. Do we need to wage wars over that while there's people around who deny temperature?
What happened to "That's, you know, your opinion, man"?