Puzzle Battle problem

That should not be the case. In your scenario where both players solved the same number of puzzles, player 2 should win since they had fewer failed puzzles.

But if he fails the next one, it is a draw.

Also in no. 1 was a typo, Player 2 wins if 2 puzzle failed

Yes, puzzle battle is made this way to reward those that get fewer puzzles wrong. If they get the next one wrong, it just means that they had the same number of puzzles that were incorrect, leading to a draw.

This might be why https://www.chess.com/news/view/chess-com-banned-by-russia 

So it's like having a win, but you can still waste it.

Is it even fair?

It's the decided tiebreak system. You have the option to claim the win earlier in this scenario.

So, do you mean that ties are not allowed in Puzzle Battle, unless a winner is completely unclear?

It's not that ties are not allowed. As mentioned, an identical performance is a draw. It's just that the tiebreak system rewards accuracy in solving puzzles. It's always been like this. https://www.chess.com/news/view/new-feature-puzzle-battle 

Thank you. Didn't know. To whoever reads this post, please don't underestimate a moderator. They are more helpful than you would have ever expected.