[wat10000]

I can accept that there is a lot of nuance on the math side that I’m completely missing, but the Turing machine side is really straightforward. A Turing machine either never stops, or it stops after a finite number of steps. If it stops, the number of steps that it runs is a finite whole number, no different from “three” in its relationship to infinity or its theoretical ability to be written down. This doesn’t depend on your mathematics, only on your Turing machine.

[bo1024]

The point is that when it "never stops", there are models of ZFC in which the "infinity" number of steps it runs for isn't considered infinity by the model, it's a made-up "nonstandard" number that is smaller than infinity but larger than any integer. And that model considers that to be "halting", so that model says the TM halts.

[wat10000]

That’s just a change of definition. That isn’t really saying that BB(748) is different under a different model, just that there’s a BB’ equivalent for that model and BB’(748) is equal to something else.