Maybe not what you are trying to discuss here, but: the "Shooting" genre has been pretty much fucked up as a name, people think of "Shooters" to FPS these days and for me the original genre that really deserves that name is the Xevious/1942/Gradius type of games. Now the genre is called in the US... "Shmups" and eeeww, honestly. On a side note, in Spain the genre is called "matamarcianos" that means "martian killers" and it's even uglier haha.
I don't like "Beat 'em up" too, I think the japanese concept of "Belt Scroll Action" is much, much better.