Question regarding insufficient material

I played two bullet games this week with for me surprising outcomes. The first one was
Here I lost on time as white against king and bishop. I noticed that a checkmate is indeed possible if I'm having a seizure and walk into the corner with my king. But then I had another game
where in my opinion regarding the first game I should have won but the result was "draw due to insufficient material". I played this later game on chess . com. My question is whether the big chess websites have different rule sets or if this game would have been draw on lichess aswell.

FIDE Laws of Chess:
"6.9 Except where one of the Articles: 5.1.a, 5.1.b, 5.2.a, 5.2.b, 5.2.c applies, if a player does not complete the prescribed number of moves in the allotted time, the game is lost by the player. However, the game is drawn, if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves. "
Game 1 black wins on time are there is a possible series of legal moves leading to checkmate.
Game 2 is also a loss on time: a series of legal moves leading to checkmate exists.

rules on draw on differnt sites. But so do they vary from country to country sometimes. FIDE rules say that assisted mate is enoughbutI think USCF requirematerial that can force a mate. But Ido notthing there is even assited mate in the latter example. Not sure though. But probably requires sufficient material that and in principle deliver mate.

Can't Black mate in the last game? Black should have won

Lichess follows FIDE rules. So if white lost on time in the 2nd game. The point is awarded to black since a checkmate can occur. follows different set of rules.

Sometimes results have to be corrected on ICC (USCF rules): server declares a draw but you have forced win - now you got to claim... (say N vs h-pawn)

So first game above where black black only has a bishop and a king, black wins on time even though he has insufficient material to checkmate? Is that's how games work here? I saw a thread elsewhere where a guy was saying he was "Banned" (I guess temporarily) for not resigning. If both are true, it's like people are getting mixed messages.

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