"hamburger" was named for Hamburg Germany where it first appeared.
Ground beef was from around 1894. Our meat grinder was invented about that time in Germany about 1864 for making sausage but might be a little older.
Hamburger was a name for any ground meat. Horse meat was also called hamburger as well to hide the fact that it might not be all beef. By the end of WW2 all ground meat had to say if it had anything but beef if sold in the US.
To this day hamburger meat must show any other products other than beef.
This includes hamburger products.