However, there is some consensus on socialism, which is the term the Nazis used. Fascism was originally linked to Mussolini, and not the Nazis.
And the book that started the conversation we entitled
Liberal Fascism: The Secret History of the American Left, From Mussolini to the Politics of Meaning
So why are we debating the Nazi's?
But the point you made about not defining terms is all too common, and a pox on our national discourse. Same with "liberal", "conservative", "right", etc. When the dull witted Hannity had Michael Moore on to debate his latest hit piece of a documentary, they never defined the word "capitalist". Michael Moore ate his lunch, both figuratively and quite possibly literally.
In 1932 Mussolini wrote (with the help of Giovanni Gentile) and entry for the Italian Encyclopedia on the definition of fascism:
..Fascism [is] the complete opposite of Marxian Socialism
The entire article by Mussolini makes for interesting reading, I would suggest it. I makes point that support either of our propositions.