If your job was working on a hog-farm, with the task of collecting the sperm of prize-winning hogs to perpetuate the line, I'm pretty sure that you'd like to be called a "hybrid rider."

Actually I've never heard a person being identified as any sort of rider - except in print perhaps. I find 'cyclist' or 'bicycle-rider' in use. But not to the degree of specification of a particular type of bicycle.