You are correct, there was do direct connection between the two, my bad, its like comparing Green Berets to the Military Police in some regards.
It is unfortunately a common habit to refer to “Nazis” instead of “Germans”. The history of it, as has been explained to me, is that it was invented by the Soviet propaganda to in a way shield their vassal state the DDR, that is the “good Germans” (in fact there was as many if not more former Nazis in East Germany as in the West, and certainly more that were never pursued for their wartime crimes, simply because Communist states were essentially lawless) from association with Nazi Germany. The monicker has served us in the West as well, and has become common. Yet, only about 6% of the German population belonged to the party.