A persistently missing name is often a sign that the concepts haven't been fully worked out yet. In you guys' case that could be exciting. It could mean that what you're doing is newer and more emergent than you've yet realized. If so, your job is not to name the thing, but to follow it as it emerges and let it teach you what it is. Names come almost as a corollary. Until they do, lack of official names for the core concepts (what could be more core than "student" and "school"?) is an asset, not a liability.
You probably already have a laboratory running natural experiments in this area: the conversations of the people in your groups. If they lack official language for what they're doing, they have no choice but to make something up. Those can be good creative conditions. If I were you, I might try to get them to tell me what the names are, by listening closely to the words they use in conversation and to which formulations feel easiest and most alive. This often works better when people aren't consciously trying to come up with names but do it spontaneously when talking about something else.
By the way, maybe you shouldn't be down on "school" for what you're doing. It's true that people have a lot of preconceptions about school. But it's also true that your usage is historically deep. "School" didn't originally mean "child processing factory" or whatever it is now. Here's what Etymonline has for it [1]:
"place of instruction," Old English scol, from Latin schola "intermission of work, leisure for learning; learned conversation, debate; lecture; meeting place for teachers and students, place of instruction; disciples of a teacher, body of followers, sect," from Greek skhole "spare time, leisure, rest ease; idleness; that in which leisure is employed; learned discussion;"
That seems pretty close to what you're doing. Historical meanings often have resonance and sometimes make comebacks. I wouldn't relinquish this one if I were you. Imagine if you are part of a resurgence of the original meaning of schools. How cool would that be?!
[1] http://www.etymonline.com/index.php?term=school
p.s. You might consider "mentor" for what your web site calls "resident", and then "resident" instead of "student"...I think "resident" could go either way.