If not, what word was used by the Germans themselves during WW2 for "a member of NSDAP"?
In short, the Third Reich never almost never (see below) used the word "Nazi" in its official lexicon and its use was actively discouraged by the state. The term itself was a derogatory term with its origins prior to Hitler's seizure of power. Many NSDAP members often referred to themselves as NaSos (short for National Socialists), and their opponents instead called them Nazis. The word Nazi had a negative connotation in southern Germany because it resembled the Ignaz, which the rough translation is "ignorant" or "blockhead." The word Nazi became quite commonplace among the many political enemies of the NSDAP, more than a few of which ended up in exile after Hitler's rise to power. These emigres would help popularize the word Nazi as they often wrote disparagingly about the direction Germany had taken since 1933.
German written discourse in the Third Reich had a number of means to refer to both the National Socialist political movement and its ideology. Most commonly, writers tended to use "National Socialisten" or "National Socialistische" as either a noun or adjective. Writers would sometimes also use the abbreviation N.S. for space saving purposes (this is quite common when referring to state organizations such as the NS-Frauenschaft (National Socialist Women's League). For the Party itself, the acronym NSDAP was pretty universally used.
To add a little more to the importance of names, during wartime the Soviets frequently referred to the Germans as fascists. This was in part due to the Marxist-Leninist idea that Mussolini and Hitler were part of the same counter-attack by capital on the rise of the USSR, but also a means to not call attention to the Socialist component of the NSDAP's name. The label "fascist" struck a number of German observers as somewhat odd because the Third Reich made it rather clear that the NSDAP was a distinct political movement quite separate from Mussolini's one. While Germany's state media stressed that Germany and Italy shared common interests, they were not part of a single political ideology and the word "fascist" was mostly used to refer to affairs in Italy.