To be sure, all were forced to swear the oath of allegiance to Hitler (Eidgeloebnis), making them, yes in effect, "Nazis"!
All are correct in a way.
After '33 the NSDAP imposed an admission freeze for new members, to avoid all these opportunists who now came running.
At that point the party had about 849.009 members. Of course even after the freeze people seen as especially useful were "rewarded" with a membership.
But membership was in no way compulsory like in communist countries. (Quite the contrary, the lower your membership-number the higher would be your reputation, because that meant you had been an early member as it was still difficult, during the struggles, before the rise to power).
Till 1943 about 11% of the Germans were members....actually really NAZIS as in active members of the NSDAP. Meaning 89% were not.
But swearing allegiance to Hitler (not the party or the country) with Hitler being the party leader meant in real life actually the same...