I take my definition directly from Hermann Goering who was a pretty good authority. Extreme nationalism is an essential element. See Italy’s Mussolini, Argentina’s Peron and Spain’s Franco. We use the term too frequently to describe any authoritarian figure. Putin might be seen as a fascist today.
So he wasn't at all interested in the idea of racial purity, and fascism to him was primary an economic system where the state is supreme yet within the context of some level of private ownership of industry (i.e. corporations). Well, he did say this:
Now of course Mussolini was a huge nationalist, a subscriber to the Italian irredntism movement. Recall that Italy was an infant nation cobbled together from previously autonomous states. Just a few decades before there was no such think as "Italy". Hence Nationalism in this context was to say that Italy as a state was legitimate and powerful. Viewed through this lens, the nationalism of Mussolini and Franco were both driven by the circumstances of their specific situations: new nations that were still being established. Nationalism then is an appeal to unity where one might otherwise see the state fracture. It made sense if you wanted to be a dictator of a nation that you must first support the idea that there was in fact ONE nation here to rule.
But does that makes sense in the case of established states like, say, China or (should it rise here) the USA or France? Would an appeal to national unity be necessary to consolidate power as it was in Italy and Spain? I say no. If you have all the other characteristics but not the overt nationalism, you have a fascist state.