Actually shooting is like a part of Action but not fully. It's like a section. You know when you pick something, you get more options for it? This is just like it.
First of all, threads that require a yes or no answer only are considered spam. You need to encourage more substantial responses.
As far as separating these two categories, they both make quite large categories on their own. Some of them can fit both category, but some shooting games are not filled with action.