can you use the engine of d&d to not play d&d
Mutants and Masterminds. So, yes. Mind you, that is a havily modified engine. But also mind that most every homebrewer modifies the engine somewhat....
can you use an engine that is not d&d to play d&d
The last time I played Old School Hack... it felt pretty much like D&D to me. So, I'd say probably yes.
what is the distinction? Are some people just using the words d&d to describe any rpg?
For the second, probably not. "D&D" is not being used as "Kleenex" here.
I think the ambiguity of language is a useful thing. There is "Playing D&D" - using a ruleset published by an appropriate copyright holder
as D&D. Then there's "playing D&D", which is playing a game that has enough of the traits of D&D as to feel as if it is in the same sub-genre as D&D itself.
And, much of the issue is right in that word - sub-genre. Folks like sharp lines on their definitions, but genres aren't like that. If you have enough of the tropes (which include, but are not limited to, the published mechanics) then you're in the genre. Genres are defined by inclusion, rather than exclusion.
is asking these questions gatekeeping?
Asking questions when you don't have a presupposed answer is not gatekeeping.