You would deal the damage to the other player's hero card. The target of the damage is specified, so you don't get to choose other options in this case.
The defender in a fight is the opponent of the player who starts the fight. (
https://munchkinccg.game/gameplay/rules/#Start_a_Fight )
And you are correct, player and hero are largely interchangable when referenced. (
https://munchkinccg.game/gameplay/rules/#Hero )