Emeton keystring just a detail, Just Intonation is not a temperament at all. It’s a technique that players use while playing with others to raise or lower there pitch frequency to achieve a certain harmonic quality. It’s not one of the temperaments.
I had to check back. I did not use the term "just intonation" at all, but when responding to GaryD I did state that his reference to equal temperament went to something that you had written - and in your post you did indeed refer to just intonation. My own exposure was two decades ago before I ever studied theory. There was the idea of tunings on violin, and one of them was "like the piano" / referred to as equal temperament, and the other wasn't.
For learners trying to get a basic idea of the naming system of intervals, I don't think we need go to tunings - esp. for piano students since we have pre-tuned systems. On piano if we play C Eb and then C D#, we have exactly the same quality. It's simplistic but it's a start.
For the other tunings, it starts getting sticky when other instruments are involved - for example, do you want to clash with a piano if you want your D# to be slightly flatter that Eb but the piano will still have that same tuning since it's a piano? But that is way past the scope of the question of naming intervals for piano students and might want its own thread for those interested.
