I agree that "rough landing" is more common. It is a general term for any landing that has problems. The opposite is a "smooth landing", like almost all airliner landings.
I think "hard landing" and "soft landing" are terms mostly used by pilots, with a very specific meaning.
If you "land hard", your wheels hit the runway with a lot of force (the airplane is moving downward too rapidly when it touches the ground). In a "soft landing" your wheels hit the runway sofly (the airplane is moving downward slowly when it touches the ground).
I really think that "hard" is acceptable, but I think it is focused on as dojibear said the actual impact. I would expect "bumpy" for example to refer more to air turbulence and the airplane moving around a lot before actual touchdown. I think the term "rough" is actually inclusive of both possibilities. "The landing was pretty rough" to me could indicated either a bumpy turbulent descent before the touchdown and/or a hard impact on touchdown.
A hard landings is one where the wheels strike the runway with a jolt.
A rough landing (in my mind, at least) is one that involves more than just that one jolt.
I did find a definition for hard landing, when I Google "rough landing" I get sent to the same "hard landing" definitions. I suspect that "hard landing" is a recognized aeronautical term and "rough landing" is only a civilian term: