Where exactly is the gatekeeping ?
The gatekeeping is in the distinct implication that people that are not, in fact, role playing.
Please go and have a look at the definitions of the rulebooks for every edition. I have, and these are the ones I employ, since this thread is not only about 5e. None of these definitions are mine.
You specifically said you use the ones in the book - you adopt them as the ones you are using, so they are yours at least to that degree.
But, honestly, what definitions you use for yourself are irrelevant. You are totally in your rights (indeed, in your responsibilities) to set expectations for your own table. That should not be a point of argument here. The issue is the pressure/judgement that somehow these are the correct definitions, and others are lacking.
The OP, in fact, doesn't go into what the definitions
should be. This is smart. We should be able to go back and look at the books, and see what the definitions were there, and speak also about what the actual definitions used in practice were - we always note how a great many original rules were ignored at the table, after all. One of the big takeaways from the discussion here is that the practical definition differed from the printed one. We could (and have somewhat) engage in discussion of why there's a difference.
We should not feel a need to tell anyone, "Well, what you are doing now is not even roleplaying - look at this definition and tell me I'm wrong!"