Reflecting on this, I like anarchism as a negative term. It's like breath meditation. It's so easy to get distracted and get lost in the goings-on of the day that you need raw negation to reset and notice things with fresh eyes. Likewise, it's so easy to defer and get lost in the politicking of a society that you need raw negation to reset and notice things about the politics. And that negation needs to be a regular practice if you want to maintain it.

So specific forms of anarchy such as anarchocommunism rightfully places the negation first. You have to keep stepping away from the structures and see whether they are doing what you want, and if (your perception of) communism doesn't have the answers then you leave communism behind.

I could see 'horizontalism' being warped into a deference to peers - a lynch mob is a horizontal structure, after all. I could see it taken as a prescription to wipe out inequalities like talent or disability. I don't know if these are fair, but I do know that if horizontalism became the dominant term then 100 years from now there will be people making these complaints. Because it is the nature of words to absorb the felt meaning of what they represent. There is a long list of words for mental disability that are now slurs or slurs old enough to be mere insults because people in the English-speaking world and elsewhere have been trained to feel revulsion at (mental) inability, and no matter how carefully a word is chosen it will absorb that vile intent.

So much of the unwellness of the term is in societal perception rather than the term itself, and will travel along with any rebranding. I do acknowledge your point of white supremacy in western anarchist philosophical discourse, and if a change in name helps reduce that white supremacy even temporarily, then good.

However, I don't think one person behind a desk imposing a name on dozens of non-white ideological currents based on a western anglophone vibes-based analysis of their supposed shared concepts is a good starting point for reducing white supremacy or dismantling colonial attitudes towards non-white cultures and philosophies. If these currents even have any interest in being grouped together into a single worldwide term, then let them name themselves.