In the bare King case counting is automatic. It does not have to be announced or done, and after the prescribed number of moves have been played the bared player can simply claim the draw.
In the other case any player that could still theoretically lose can start counting if the opponent is not yet counting.
Note that even though there are no checkmate positions in KN vs K, there are checkmate positions in KN vs KR. There are also checkmate positions in KNN vs K. Would such positions still be draw if the strong player forfeits on time?
[Edit] I still have one more question: Suppose a game has reached a position with KBBP vs KBN, but the black B is trapped against an edge, and cannot escape being captured. But black is on move and captures the white Pawn, with the aim to start a count.
If both players want to count now, who gets the right to do it? If I were white, I would announce a count in the position KBB vs KBN, and count my move KxB as 7, since there are 6 pieces before that move. And with KBN, as the opponent had before the move, he can still force a win. What is my ultimate count now? Is it 22, because the counting player has two Bishops, or 44, because the opponent (the non-counting player) as one?
In any case, I would happily continue counting in KBB vs KN. Now this end-game contains some forced wins, which typically take 32 moves (maximally even 45 moves),. Suppose I am in such a theoretically won position that would take 25 moves. After my count reaches 21, (so 14 moves later) and I am still 11 moves removed from achieving a checkmate, I announce that I stop counting, and play the move. Now the opponent can start a count. But he has to start at 6, and the ultimate count is now 22, because I have two Bishops. So he has to survive for 14 moves to claim a draw. But it was a mate in 11, so he gets checkmated before he can claim a draw.
So in total, I have forced checkmate in 25 moves. If my opponent would have started a count immediately after he captured the Pawn, I would have gotten only 22 - 7 = 15 moves to checkmate him, and he would have claimed the draw 10 moves before the checkmate.
So both players have a good reason for wanting to count in this case: black to get a draw before he gets mated, and white just for preventing black from starting a count until it is too late. Which player gets priority here?
OK, so if I understand you correctly:
Note that even though there are no checkmate positions in KN vs K, there are checkmate positions in KN vs KR. There are also checkmate positions in KNN vs K. Would such positions still be draw if the strong player forfeits on time?
[Edit] I still have one more question: Suppose a game has reached a position with KBBP vs KBN, but the black B is trapped against an edge, and cannot escape being captured. But black is on move and captures the white Pawn, with the aim to start a count.
If both players want to count now, who gets the right to do it? If I were white, I would announce a count in the position KBB vs KBN, and count my move KxB as 7, since there are 6 pieces before that move. And with KBN, as the opponent had before the move, he can still force a win. What is my ultimate count now? Is it 22, because the counting player has two Bishops, or 44, because the opponent (the non-counting player) as one?
In any case, I would happily continue counting in KBB vs KN. Now this end-game contains some forced wins, which typically take 32 moves (maximally even 45 moves),. Suppose I am in such a theoretically won position that would take 25 moves. After my count reaches 21, (so 14 moves later) and I am still 11 moves removed from achieving a checkmate, I announce that I stop counting, and play the move. Now the opponent can start a count. But he has to start at 6, and the ultimate count is now 22, because I have two Bishops. So he has to survive for 14 moves to claim a draw. But it was a mate in 11, so he gets checkmated before he can claim a draw.
So in total, I have forced checkmate in 25 moves. If my opponent would have started a count immediately after he captured the Pawn, I would have gotten only 22 - 7 = 15 moves to checkmate him, and he would have claimed the draw 10 moves before the checkmate.
So both players have a good reason for wanting to count in this case: black to get a draw before he gets mated, and white just for preventing black from starting a count until it is too late. Which player gets priority here?