Side note: both are percussion majors and not used to reading music per se.
Mod edit: I have moved this from the Piano Forum because it doesn't deal with pianos as such. FB
I was a music major and took five theory courses and I don't remember if choosing either "half step" or "semitone" was problematic. The only time such terminology really helped me was in the memorization of diatonic scales. It's easier to remember a major scale, for instance, by thinking of whole and half steps, like WWHWWWH.
Side note: I am that old, yes, my final National Piano Guild exams where the Van Cliburn Competition, and still feel it is a great loss to the Guild , not to be aligned with that Competition.
Thank you again for your supportive reply, Oberon
I know it's very eccentric, but I myself usually think of harmony in cents, personally having a better head for fluidly juggling numbers than a combination of words and numbers) and dabbling with microtones (where the application of words for specific intervals/harmonies can get incredibly cumbersome and, at times, arbitrary, and the use of ratios, while often giving a sense of where in the overtone series a pitch is, is difficult to visualize in actual 'pitchspace').
Quantative: Number of tones/semitones/whole-steps/half-steps between two notes
Qualitative: The interval between two notes in terms of the harmonic or melodic quality of that interval, ie: minor second, major second, minor third, major third, perfect fourth, augmented fourth/diminished fifth, perfect fifth, minor sixth, major sixth, minor seventh, major seventh, octave, minor ninth, etc.
I think that both have their uses. I taught myself theory as a guitarist and as a pianist, so being able to measure the distance between two notes in terms of frets or keys was always useful. Analysing music however, I tend to think in terms of the name of the interval. I think that ultimately it's useful to be able to think and speak using either system. I think that all this terminology is only useful as a means of communication, and being able to quickly understand what someone else is talking about empowers you. Similarly, being able to talk about intervals the same way as the person you are talking to will make it infinitely easier to make them understand what you mean.
For teaching, I suppose encouraging both systems is worthwhile, provided it's done in such a way as to not cause confusion. I advocate using both systems simultaneously so that the pupil is aware of how one system relates to the other. The danger might be that both systems are taught individually, but at the same time, which will confuse a student, leaving them unsure about how to discuss these things, hampering their progress.