What is considered insufficient material?

In your example that's simply checkmate, clock is irrelevant.  Insufficient material is only determined as such(and therefore a draw) when the player with the advantage is flagged.  Although having 2 Knights + King against King is only possible to mate if your opponent helps you out, it would still count as a win if your opponent flags simply because the checkmate is a theoretical possibility.  The only way its a draw is when you flag and your opponent has insufficient material to mate regardless of theoretical possibilities (when they only have a king, or only a king and one minor piece, or the inverse as well.)  Although this is my drunk explanation I think it is correct.  

I think the OP is asking what happens if white flags in the original position in the diagram (without playing a7).

Well I think the result is the same, because all the moves are forced, no deviations are possible.  

Under FIDE rules, if there is a possible mate of any sequence, the flagged player loses. Under USCF rules, if the flagged player's opponent has material insufficient by itself and there is no forced mate, it's a draw.

Yes, he can.

The first thing you need to know about that position is although black is able to deliver a mate, he cannot force mate with only a king and knight. Which brings up the second thing....

That position is actually called a help mate. Black cannot force a mate, but white assists black with the mate by walking right into it.

With the white pawn on a6 and the white king down near the pawn on the "b" or "c" file, there is no way for black to stop the pawn from advancing while also forcing white into the corner where he can then deliver mate after the unfortunate a7. White would have to make the horrible mistake of moving his king to a8.

If that did happen however, white could simply run the clock down and draw. But anyone who would make a move like Ka8 in that position probably wouldn't realize that he can simply let the clock expire for a draw. 

What's more likely is that white will just suffer endless checks until black eventually gives up and captures the pawn.

Let's suppose you are playing a game of bullet as white where you have taken all of blacks pieces except for three pawns. With the white pieces, you have a rook, knight and bishop (which is more than enough material to give checkmate), but you are running out of time to figure out how to do it.

To guarantee a draw at worst, you run your pieces acround the board and capture blacks remaining three pawns. Now black is left with only his king.

If black avoids mate and run down your clock, the game is a draw. If black had all the time in the world, there is no way for him to give mate having only a king. That's why when he runs down your clock he cannot claim the win. It happens all the time in bullet. One guy runs around the board, captures all of his opponents remaining pieces and simply lets the clock run expire for a draw.

Also notice in your position that if white did not have his lone pawn, the game would be drawn regardless if the white king was trapped in the corner or not. There is insufficient material to mate. White cannot force nor deliver a mate with only two kings and a knight on the board. 

That's the difference between sufficient material and insufficient material. If the player can force mate with his pieces, it's sufficient. If he cannot, it's insufficient. 

No.

a7 is the only legal move. Nc7# is not. Black might legally retreat his king and allow White to promote and checkmate Black.

If Black ran out of time before making the move Nc7, he should lose.

Does Black also lose if Black runs out of time having touched the knight?

Thanks everyone!

Well yeah black can choose to not deliver mate, if he doesn't realize he's playing chess.  My point was that white is forced to play a7.  But if black didn't play Nc7# (not sure why we're even considering this) then of course white could win on time, he has a theoretical possibility of delivering mate by pawn promotion.  

You mean only a fool would get into Stamma´s mate?

How about this position?

USCF and FIDE dispute...

White is not forced to play a7. He may run the clock down for a draw.

Maybe according to the stupid USCF rules (which I didn't realize).  But everyone knows USCF ratings are basically meaningless.  

It's not about USCF rules. It's the standard rules of chess internationally. What's so stupid about it anyway?? Black has no material to mate. White runs down the clock.

Happens all the time in bullet. Literally all the time.

A position is a draw by insufficient material if and only if one of the following two conditions hold:

(1) There are only Kings and one Knight on the board.

(2) There are only Kings and any number (including 0) of same-color-square Bishops (the color of the Bishops themselves is completely irrelevant) on the board.

I can provide a proof if you would like.

Look at post #5.  There is a mating sequence possible, therefore if white flags in that position he would lose.  

That's a different situation. In that case, both players have sufficient material to mate. They both have a pawn.

Where does black have a pawn?  

I don't see that anywhere in the FIDE rules. Check for yourself.

https://www.fide.com/fide/handbook.html?id=171&view=article

The USCF has a couple of rules in regards to this.

One is Insufficient material to continue.

K vs K, K vs K with N or B, K+B vs K+B if both bishops are the same color, and no legal moves that could lead to checkmate by the opponent (so if there is a helpmate, it isn't insufficient).

For a win on time, the insufficient material for the player with time on the clock is a lone king, K plus B or N and no forced win, K + N + N if the player with no time does not have a pawn and there is no forced win.

So by USCF (14 E) rules, the position in post 1 is a win for black if white lets the time run out.

FIDE is the same (6.9 covers flag fall). Chess.com is a draw, unless something recently changed.