I'll get the ball rolling with my answers:
1) Is there a difference between "key of ____" and the major scale?
Yes, there is a difference. The major scale is made up of notes using a certain formula. However, the key refers to the organization of the chords.
For example, you may look at a piece of sheet music and see 0 sharps or flats as the key signature and automatically think its the key of C. However, you could be wrong because it could be A minor, D dorian, E phygian, or even a completely different key like G with accidentals written everywhere.
2) If so, should musicians know about it?
Yes musicians should know about it. I think if more musicians knew about this, they wouldn't be in such a despair when their playing sounds bland. I hear this question get asked a lot:
"I've practiced the major scale, I've practiced basic chords, and I've practiced progressions in every key, and my playing still sucks. Why?"
There's more to a key than the major scale. The major scale in my opinion is just the starting point, and I don't think this principle is being stressed enough.
3) How should it be taught so that confusion is minimized?
That's a good question that even I don't have a full-proof answer for. However, I have made posts dealing with this, but maybe some videos along with those posts might help.