I do feel like a characteristic of a nazi does involve hate for Jews and an advocation for their extermination.
I don't mean to interrupt your discussion with the OP, just to reply on one detail.
One can define "nazi" as "someone who agrees with the proposal in Mein Kampf". Then a degree of Jew-hating is involved. But this does not have to be the only definition for a nazi. The same way we don't define anymore the political Left and Right by the seating order at the French National Assembly from 1789, or Liberalism by "someone who agrees with Adam Smith". We use updated criteria that fit today's society.
Historical Nazis hated Jews, but also hated other minorities present in Germany in the 1920s. They sought to exterminate the Disabled already in 1924, a decade before conceiving the same idea about Jews. They also committed genocide against the Romani (Gypsies). They did focus on Jews, I believe, because Jews represented the largest of their targeted minorities.
I think that if Nazism would be born for first time today, it's possible it would not focus on Jews, because Jews aren't much a topic of national importance in today's Germany. Because the minority du jour in Germany is Muslims and Refugees, a new Nazism would seek persecution of those.
Their broader definition has to do with race superiority, brutal hate for minorities, and seeking purifying a country (through genocide) from people of different races, religions, cultures.
(I don't have an opinion and am not trying to imply anything about Elon.)