The diaphragm is a sheet of tissue that separates the heart and lungs from the internal organs - the stomach, intestines, etc. By itself, it doesn't do very much. I would honestly prefer to leave the diaphragm out of the discussion altogether, since nobody can actually visualize or feel their diaphragm, but the term is so established in teaching that some discussion is unavoidable. Rather than dwell on the diaphragm, I prefer to tell students to breathe as deeply as possible - all the way to the bottom of the lungs. The whole point of breathing in properly, after all, is to be able to blow out with power and control.
Support is nothing more than pushing in with the abdominal muscles.
It is just as useless to breathe only with your belly as it is to breathe only with your chest. Our bathrooms are witness to the many ways this process can be diverted. The best way for the student to learn the proper sensation is to try a few of these natural things and observe how they feel, then duplicate the feeling while playing. A really deep satisfying yawn. Try a few more things.