HOW TO PLAY PUZZLE BATTLE?
The positions are pre-arranged puzzles and the opponent in your puzzle is a bot, like you thought. When you make a wrong move, the engine tells you. 3 wrong moves and you are out. Both you and your opponent are solving puzzles, in the same format as regular puzzles. Whoever solves more puzzles in 3 minutes wins
Chess.com has chess puzzles, where the aim is to get a sequence of the best moves. These puzzles are rated, based in how quickly and how often players solve them. You can look at puzzles I'm their raw form by clicking on "puzzles".
In a puzzle battle, both players are given three minutes to solve as many puzzles as possible. The puzzles start at a rating of 100 and go up in difficulty by 50 each time. As you correctly identified, three strikes and you're out, so you have to balance solving puzzles quickly with the risk of an error. You might want to alter your strategy depending on how your opponent is doing: if they're ahead you might have to take more risks to catch up.
Who decides what the correct answer is? Stockfish. However, don't worry; because when creating puzzles chess.com have rules to make sure you won't get penalised for a good-but-not-perfect move. Chess.com has a filtering system that only publishes puzzles where the moves of both sides are the only good move.
Of course, what qualifies as the "only good move" will depend on how good at chess you are, so I imagine the definition gets looser ad the puzzles get harder, but I'm not sure.
In the early puzzles in puzzle battle, usually getting the puzzle wrong means you've missed a backrack mate or a free hanging piece; and also in a lot of puzzles you're opponent is one move from checkmate if you mess it up; so yes, the wrong move is clearly the worst one in those scenarios.