It makes sense to present modes both ways, with the pitch set fixed and with the tonal centre fixed. That's what I do. People who play diatonic instruments (melodeon, moothie, bagpipe, clarsach) have to think in the fixed-pitch-set model.

If you really want to try the experiment of playing the same tune in different modes, you can use any ABC player (or ABC MIDI-fier) and change the "K" line in a tune to each mode in turn. What you will typically find is that one sounds right, one sounds sort-of-okay and the rest sound crap. This is telling you that modes determine melodic structures. The next step is to listen to a whole lot of tunes in various modes that all sound good as written (which is why I included so many examples).