I found an incorrect puzzle--- how often does this happen?

One idea for White would be to make the game last as long as possible. In the line I gave (up to 46. Bg3), Stockfish 13 thinks Black can expect to queen the pawn on move 53. I'll let it run with your line and see what it says.

In your puzzle after 46. Kc3 h2, White can't keep Black from promoting on move 49. So it might be a slightly tougher defense for White to bring the bishop to f2 instead of b8. But I think there's only a tiny difference, if any, between the two. I don't know whether there's any time constraint connected to solving the problem.

Yeah my puzzle is over after move 45. Objectively by bringing the bishop to f2 white prolongs the loss; the problem is it leads to positions where Black has multiple wins, which would also have to be the end of the puzzle, because there can only be one solution. Well, under the constraints of this website. There are plenty of good puzzles out there with multiple solutions, but chess.com does not allow for those.

Well, you could get one more only move out of your puzzle with 46. Bxe5.

Oh yeah! Not exactly hard to find, but hey, it makes the solution longer, lol

After hearing averagepatzer540's explanation of the unconventional nature of the chess.com puzzles, it has occurred to me that the move objected to is not the reason the puzzle is incorrect. The task of the person choosing the white moves is not to win or draw, but to present the solver (Black) with a single winning move at every turn. After Nc5, the puzzle is already incorrect, since Black has no winning moves instead of one. But by playing Bxc5, Black enables White to save the puzzle by playing Bxc5, which gives Black the single winning move h4. Now 45. e4+? is mistake, since it turns the position back into a draw. Instead the move given, 45. Kd3!, requires Black to find the one winning move, h3. From this point, each move by Black is an only move up to the promotion.

For someone used to working on ordinary studies, the logic of this kind of puzzle is a little hard to get used to.

Yeah if they just started the puzzle after white's 45th move, or even the 44th move, where 44...h4 is the only move that doesn't lose for Black, it would be ok. The problem that it starts on move 43, because there there is more than one drawing move for Black, and no winning moves.