I've always called it the Holy Roman Empire, because it was tied around the faith of the "Holy" Catholic Church. Of course, then Martin Luther came along and the empire kind of got broke up.
The reason it was originally called the Holy Roman Empire was after Rome fell everything was in chaos and people just wanted to go back to the good old times of unity, and so when Charlemagne united the tribes of Europe they had peace like when they had Rome, which was why the Pope gave him the title Holy Roman Empire
(Also, it helped the kings to make it seem like they were part of a government instituted by God and the Pope, and the word Rome had some kick in it as well)
But no I haven't heard it called Holy Roman Empire of the German Nation ever in the USA, but I was also homeschooled and have no clue what people are taught elsewhere