The phrase "green pepper" always refers to type (2). It is the name of this type of pepper (in the same way that "bluebird" is the name of a specific species of bird.)
Type (1) looks like (the mass noun) pepper, where this particular instance happens to be green. A menu should never use "green pepper" to refer to this, because it will confuse the customers.
Type (3) looks like a hot pepper of some sort. I can't identify the species. It would be called a pepper, but never a "green pepper." If the listener was confused, I might say, "I have a pepper. It happens to be green. It is not a green pepper."
If a menu advertises "pepper" (mass noun) it means a ground up version of the thing in (1). If a menu advertises "peppers" (without the adjective green, red, jalapeno, banana, ...) then you will get something like (2) or (3), but you can't really know ahead of time.
So "I like my eggs with pepper." (I want my eggs with a ground up version of (1) on them.)
"I like my eggs with peppers." (I want my eggs mixed together with a sliced up version of (2) or (3) or maybe some other type of pepper.)
For the record, I personally don't like item (2) and was disappointed once when I ordered "pepper steak" from a Chinese restaurant. I expected a steak covered in item (1). What I got was sliced steak cooked with item (2). I think that dish should instead have been called "steak with peppers" or "steak with green peppers."