Rather, the issue here is that "if you call this X, then it MUST only come from this process".
As I have tried to say before, I think you are the one who mistakes a change in label as a claim of actual physics here. This sentence in the current context is
“If you call photons from nuclear reactions Y, and all other photons X, then Y must only come from nuclear processes.”
Which obviously is true.
Common definition of Gamma Ray is actually "origination form a nucleus i.e. nuclear), which included radioactive decays.
Well, I think you should also acknowledge that this definition if of little use in other fields, where you either don’t precisely know or do not care what the source of radiation is, but just that it is around and has a certain energy. There is not one “common” definition in general, and neither of the definitions is really “better” in general. But insisting on a particular definition in the wrong context will not get you anywhere.