Checkmate or foolsmate?

This is the same game as checkmate or stalemate:

Both. Foolsmate becausesame concept as checkmate, and checkmate because...well... you know.

Why would fool's mate not be checkmate?

A stalemate is when the king has no legal moves, but is not in check. Also of course the one with stuck king can't have any other movable pieces.

Again?

 Very well. Than is what's below checkmate or foolsmate?

Because fools don't know how to write a check.

This could also be considered a form of "distant" Smother ++.

lol... I know what you're talking about. I saw your Scholar's Mate thread, which seems to be attracting quite a bit of attention.

So, if the OP's diagram is a smothered mate, are back-rank mates also smothers? And how about this one?

One of the 8 squares around the king is occupied by a black piece; however, if the piece weren't there, the position would still be a checkmate.

Well, it seems that you've already answered by first question in your thread - yes, a back rank is a smothered mate. I couldn't find an answer to my second question though.