Why move King to h1 and not to f1?

Well the two moves seems to be equally good.

Thanks for embedding it. I did not see how to do this with my smartphone.

To me both moves seem equally good, too. So it must have a single best move I assume. But this was a puzzle in chess.com (Puzzle Rush), and I did the false step. Afterwards, at the analysis board, I did not understand what is the disadvantage of the move...

I think you shared a different FEN than the one in the puzzle. Right? (Rush, puzzle)

This puzzle should be rated 4000. The lines are long and complicated.

In practice, none of that happens. The puzzle ends after 2. Qg6+, where black should play Ke5 and everything is still uncertain.

Just tested, but my FEN was correct. Do not know why the embessing of Toldsted leads to another output.

And definitly more than 1308. Also the quote of 49,1% leads to the conclusion of whether one chose f1 and h1 would have been pretty random