By most current definitions, if a person claims not to be trans they are not trans and by default that means they are cis, regardless of other things; what you would probably mean is something like a person that obtains a gender affirming care not matching their gender assigned at birth who also objects to being called trans, and obvious cases when cis people do that are eg prescribed usage of puberty blockers to slow down puberty of cis kids, or breast reduction for purely aesthetic reasons. As such, the care received is distinct from identity (today we broadly define trans as people who do not feel comfortable with gender role/expression/identity assigned at birth).
There is a gray area of people that are doubting, exploring, fluid, and non-binary and will call themselves "not cis but not sure trans applies" and that's also fine and follows from contextual usage of these terms. A possible individual's explanation: a non-binary person that presents more like the opposite sex than assigned at birth (therefore matching the above definition) but otherwise not sharing social experience with most trans people so they don't want to be called trans even though they are not cis by the above definition. This is however much more specific, individual, and nuanced discussion that happens around the tweets.
The key here is, many words have multiple very contextual definitions and usage patterns and are deeply anchored to different social norms. That's why there's so much emotions around questions like "what does being a woman means?" even in purely cis feminist world (it also comes close to other linguistic/philosophical discussions, like Wittgenstein's "game" - can you clearly define "a game" and contrast it with say "a toy" to make these distinct? What about "a sport"? Vegetables as more taxonomical concept and more culinary concept?) That's why we ask for precise language: I may be non trans but receiving HRT. I may be a woman legally and experience both misogyny AND privilege from presenting "manly" and having "manly" professional image.
Some people feel like this needlessly muddles things because they claim "they know what the words mean and there's one definition and languages do not change unless dictionary publishes a new definition", but honestly, they either have not thought things through, not met enough diverse people, or follow a bigoted agenda".
But again, this is too nuanced for the tweets; the simple answer is "unless there's a specific other context that forces a different definition, you either claim to be trans, or you're cis".
Similarly, one cannot be "straight" without acknowledging existence of same-sex attraction, so people uncomfortable with that won't describe themselves and expect everyone to assume they're "normal". These people don't teach kids to say "they are straight", they prevent them from obtaining any knowledge about human sexuality at all to avoid even the possibility of admitting "exceptions" exist