Yes, that’s what “Englishes” is supposed to mean, roughly speaking

Oh I’m not doubting the existence of the term “Englishes”, it’s just that I’m quite sure I only speak one of them. I can do CS technobabble and go for a more academic register, I can be colloquial, but overall it’s still the same, broad, variety. I can also do the same in German and it’s all Northern Standard German (Missingsch).

This is about a many-one vs. many-many relationship. Compare “Looking at other players’ head” vs. “Looking at other players’ hands”: Both times we’re talking about multiple heads and multiple hands, but in the first case you have a head per person, in the second you have multiple hands per person. If it was “other players’ hand” you’re probably talking about a game of cards. Though TBH this is quite awkward because idiomatically it’s “looking at the head of other players”, how dare you use an actual genitive.

Be the reason why you used the plural as it may, it’s exactly these kinds of semantic details and edge cases where language evolves easily because you can say either “hand” or “hands” and be understood because people have the context that you’re talking about poker or gloves, context which takes precedence over your choice of plural. Within your poker or knitting club, as the case may be, a standard will emerge and you have a little, baby, micro-variety of English. If it matches up with what other clubs are producing then the variety grows and stabilises.

Not at all linguists are convinced that Euro-English can be classed as a variety, true, but that’s mostly because there’s no proper definition of a language variety when used as a Lingua Franca, all the definitions linguists have assume native usage. Pretty much all of the discussion is about “do we want to define this such or not”, not “are there things which can be identified as typically Euro-English”. It’s an identifiable thing, what linguists are arguing about is whether it’s a variety.

And, again, I have to emphasise language proficiency of many speakers who use these identifiable forms: C-level speakers have access to poetic registers, language at its most “anarchy is order” stage. To class much of what they do as “mistake” is akin to classing Yoda speak as “broken English”: Very much not it is, poetic register it uses, many a renowned writer using the construction you can witness. A non-native speaker preferring that construction is not a mistake, it’s an idiomatic preference (Yoda doesn’t always use OSV, btw).

and, interestingly enough, noting that some of them appear to be arising or at least acceptable in native English too

That’s not terribly uncommon. Drastic example: Drop a native English speaker into Germany and within a couple of months they’re going to catch themselves saying “handy” instead of “mobile” when speaking English… and then fight it tooth and nail. But “beamer” for “projector” will fly straight past their language integrity sensors, presumably due to lack of sexual implication. Humans are funny like that.

Immersion aside languages, especially related languages, can have common evolutionary directions. E.g. Low Saxon and English both lost the ge- prefix in past participles and simplified case structure the exact same way after splitting into different languages. Low Saxon lags behind when it comes to getting rid of noun classes but the evolutionary direction is definitely there¹. A change affecting all English varieties could conceptually very well start in Euro-English as one of the erm “source” languages of Euro-English could be further ahead in a shared evolutionary direction, accelerating the process of the change happening in Euro-English, and then finally the other varieties saying “oh, that’s neat, why didn’t I think of that”.

¹ Low Saxon would use “Looking at the players their hands”, btw, both for a many-one and many-many relationship. Decide for yourself how well that works in English.