You both forget that this is a european subreddit. Not all countries have the same definition for words like this, and although we all speak english, it's understandable that people have different definitions for words, especially for a word like "nazi".

Dutch dictionairy defines nazi solely as a nationalsocialist for example. On top of that, different dictionaries certainly will give different definitions. It seems both of you are misunderstanding each other solely based on language barrier, and are not actually having much of a political disagreement as much. It's painful to read trough :I

Anyways, in my opinion, (if it's worth anything) using the word "nazi" to describe racists or people on the far-right is leaning towards a "reductio ad hitlerum"; you are loosely associating people with the nazi party which is fallacious in nature. But then again, it's a counterprotest, not an actual political debate or conversation. It's a bit redundant to complain about insults or fallacies when you are talking about groups of people shouting at each other.