In reply to The Green Giant: If I had to choose one activity from the two - either never climb again or never go for a walk that wasn't mostly climbing - I'd have to stick with walking. To that end I'm a walker, but I'm not sure I could call it my single "primary" activity. For me, the happy medium is somewhere between the two, doing a mixture of all (rock/ice/walk/scramble/camp...).
The only word that I would use to describe it is "Mountaineer", even though I haven't done much alpine stuff and nothing in the greater ranges! Climber and Hillwalker are both inadequate terms, at least for me, since they are only partial.