>> I don't think sharing a prefix/root implies that they're the same thing.

I think the complaint is that the the shared prefix/root causes the two words to be less distinct from each other

>> For example, when exactly do you "login" when calling an API with a bearer token? Doesn't work for me.

In my mental model, you "login" to the API when you provide the bearer token.

While I would agree that this is "stretching" the meaning of the word login quite a bit, passing the bearer token serves the same functional purpose as a human keying a UID / PW combo.