Is it technically a checkmate if the king is not in check, but all moves will result in check? In other words, after white's move, black king is not in check.  It is now black's move, but any move that black could make results in check.  Is this a checkmate?  Is there a chess term for this situation?
Putting your king in check is not a legal move as you've realized.
Of course, if Black has any OTHER legal moves he can and should play one of them!
If a side TO MOVE does not have ANY legal moves, that would be a stalemate, not a checkmate (which is delivered only by the side making the check)