It is indeed semantics as our definitoon of neo-nazi seem to differ:

"Neonazism stands for the resumption and spread of National Socialist ideas in German-speaking countries after World War II and the end of the Nazi dictatorship. National Socialist ideas were also taken up in other countries after 1945, namely in the United States, and the Nazi regime was glorified. Representatives of neo-Nazism are called neo-Nazis; the term contrasts with "Altnazis", i.e. the carriers of National Socialist ideology who had already espoused it during National Socialist rule."

That the common definition I'm used to so we are indeed talking about neo-nazis. Or are you assuming those are either old enough to be actual nazis or learned their views from birth in direct line from their nazi families (to categorize them as orginal nazis) instead of part of a resurgent movement using parts of the same ideology?

PS: Also you seem to assume that far-right or neo-nazi is playing something down. It isn't. Every form of hateful far-right ideology, nazism or fascism is equally despicable. The "real" neo-nazis only get "bonus" points for also openly venerating a former genocidal regime...

The term Neo-Nazi is used for any post-war “militant, social or political movement”

If this is the quote from Wikipedia then you left out an important part:

Neo-Nazism comprises the post–World War II militant, social, and political movements that seek to revive and reinstate Nazi ideology.

Personally I think, mixing the two terms together is perfectly fine as the only major distinction is that the original Nazis were the members of the NSDAP and neo-Nazis are not. A distinction that is not really that relevant anymore especially when talking about the ideology, which is the same.