The word has been changing for a long time. For lack of a better way to put it, think of it as the difference between an insider's word and an outsiders. Older texts almost always have an implicit reference to a particular faith in it. A 16th writer who says "he is a religious man" means that "he is an observant $SECT". In modern uses, it almost always means "he believes in this class of beliefs and practices". The reason I refer to it as an outsiders term is that it groups together groups that don't generally think of themselves as one.

Modern usages of the word "religion" group Christians and Muslims (for example), groups that would see themselves as distinct.

Interestingly, you can see a bridge period of sorts. If you think back to characters in movies of the 30s and 40s saying "I am not a religious man, but..." or "I am not a praying man, but..." you can kind of see the shift. A little reference to the good standing meaning but also some of the outsider type frame.

But isn't that ingroup/outgroup-dynamic still happening? Very few westerners would consider the followers of Bagwhan 'religious', they'd rather use something like 'cultists'. Aum/Aleph is a 'cult', even though it represents itself as a syncretism between Buddhism and Christianity. Radical forms of Islam are sometimes called a cult, and sometimes a religion, depending on the speaker.

All that shift in meaning seems to do is that it determines which belief systems are considered appropriate by a speaker, and which aren't...

Agreed, I've never heard of any group self-identify as a 'cult' in modern times. The word almost always has a pejorative connotation to it.

As a funny aside, I went to look up the etymology of the word and a quotation is listed at https://www.etymonline.com/word/Cult:

"Cult is a term which, as we value exactness, we can ill do without, seeing how completely religion has lost its original signification. Fitzedward Hall, "Modern English," 1873"

It strikes me as funny because the problem with the word 'religion' is noted as far back as 1873, but I'd argue 'cult' is now even less precise than 'religion'. I don't think it was always that way. References to Roman mystery religions as 'cults' generally lack the pejorative connotation.