What if Black calls the TD during his move without making the final stalemating move, after … g2+; Kg1 ? Even though he has no good moves (stalemate, or lose his pawn), it seems to matter.
What if Black calls the TD after … g2+; Kg1, Ke4!!; Kxg2 in other words, K vs. K. Is the game “immediately over” with K vs K the same as it is for stalemate?