Both "joystick" and "gamepad" describe the hardware format.
The systems in Ubuntu as a type of joystick (/dev/input/jsn devices, jstest, jscal, etc). Older users will probably preference "joystick". Console users will probably preference "controller".
Nobody has seen fit to tag something as a game-controller to date. It's not a natural expression.
Hardcore sim users wouldn't call a joystick a game controller.
We can discount the last group but I still don't know whether we use Ubuntu's classic convention or what is most commonly used. Is picking something else ("game-controller") better than something that's already used?