I'll reiterate my stance that only someone having fun with you by being pedantic would ever bother to deny that use of the term "Nazi" is an appropriate descriptor for any member of any fascist political movement whatsoever because most people aren't going to have a deep enough knowledge of the particulars of Nazi ideology which may have distinguished them from other fascist movements. That's just not what normal people are thinking about when you use that word.
If you're writing an academic paper then knock yourself out with the ridiculous technicalities. Professionals love that ****. In casual conversation it's completely unnecessary.
"Nazi" can be anything between a very loose metaphor and something literal. A question would be if at some point in that progression which nears the literal end, the characterization gains more gravity. Eg, is someone who is just obnoxious and vaguely right-wing, a nazi? And if so, how do they compare to someone who has actually killed a member of a minority specifically due to hatred of that minority. The latter isn't typically (literally) a nazi either, but certainly the characterization would have more value.