Even though both schools may have "BYU" in their name they are in fact different schools. BYU Idaho is not a satellite campus of BYU, it is a separate institution. Calling BYU "BYU Provo" insinuates that BYU and BYU I are somehow separate campuses of the same school which they are not. I can see how folks who go to BYU I or BYU H would like to think that but it simply isn't the case.
It isn't elitist it's simply the truth. I wish the church would have stuck with Ricks and not changed the name to BYU Idaho. It just confuses people who aren't familiar with BYU and BYU I students only add to the confusion by not making a distinction between the two schools.