This is just weird nitpicking if anything. "Nazi Germany" is just a common name for that specific government and pretty much just as valid as "Third Reich".
If you want to be really technical about it, the official name for the German state in English was "German Reich" between 1871 and 1943, though often it is also translated as "German Realm" or just simply as the "German Empire".
If you want to be really technical about it, the official name for the German state in English was "German Reich" between 1871 and 1943, though often it is also translated as "German Realm" or just simply as the "German Empire".