That’s not what the entry is saying, just that opponents favored the Nazi abbreviation because it also meant a foolish person. The latter sense was derived from Ignatz, not the former.
I concur. Big difference between “He’s a nazi” and “He’s a Nazi.” Lowercase nazi could mean anything from somebody who holds views similar to those held by the Nazi party or merely somebody who is controlling and fanatical either generally or in a specific area/activity (e.g. “Don’t be such a radio nazi! I want to listen to some of my music, too, ya know!”) Capital-N Nazi means somebody who is (well, I suppose “was” is more accurate) a member of the Nationalist Socialist Worker’s party.