for the definition what a nazi is, the movie "american history x" - that imo shows exactly what a nazi is: the main character is not german and not even in germany, but shows the nazi-mindset exactly! He glorifies hitler's thinking about others, fighting for the wrong idealogy for "own" country. ... Thats making a nazi. Neglection/sympathy for holocaust, supremacy thinking, .., too.
As I was told, the word nazi was invented by U.S. for the hitler and his followers so its a short word for national socialists (the party, its memembers and sympathisants) the US was fighting ("name your enemy"). Your question is about/implies nazi as "an individual person with a special characteristic", which makes that someone a nazi.
Its a "nation first + everything for the nation" thinking. For me a nazi is someone who glorifies the nazi time, saying it was all well done then, wanting those times back - like some elderly tend to say "earlier, everything was better". They dream of powerfull country with a straight people-body. Nazi think in races. And always say others are guilty for one's own failure, argument with "they" and "us" when they talk about own country/others.
So basically, patriotism, skinheads and far right are subsets of "nazi", but not full nazi :)
And may be, if one wants to be a REAL full nazi, its a must to shave the head and to have some on topic tattoos with being caucassian at the same time. This is how we differentiate them in germany - you see: "we" and "them". But i'm not nazi :)
funny, in US there was a nazi gang a few years ago, with the group's "leader" having the room full of nazi-stuff, flags and propaganda - as it came out later, he was jewish by himself and did not tell ..
Frank Meeik, Daniel Burros, but the one i mean was just like a joke of a nazi. But still dangerous thinking.