I’ve gotten several comments deriding me referring to an Elizabeth Line station as “a tube station” but so far—despite my requests—nobody’s explained WHY.

Is it NOT considered part of the Underground?
It has the same logo but in purple, which I assumed was just for Queen reasons.

It interfaces WITH Underground stations, does it not??

Is this a colloquialism I just won’t understand?

@Graham_LRR I think simply for many "The Tube" is just The Underground proper. The Elizabeth Line is run by TfL and has a purple roundel, which makes it another of what TfL calls a mode, like Underground, DLR, Overground, Buses, etc.
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@Graham_LRR Oh, re-reading, you're actually asking if it's not Underground -- yeah, it's not; the Elizabeth line is basically the east-west version of Thameslink. Other than being operated by TfL it's the same idea -- it uses regular Network Rail owned stations outside of the central part, you can use regular train tickets on it as you found, etc.