Battle damage

Battle Damage is the damage inflicted to a player's Life Points from battle. Monster cards can still be destroyed even if Battle Damage is not inflicted. Damage from direct attacks count as Battle Damage.

0 battle damage, such as an attack from "Mucus Yolk" with 0 ATK does not cause damage, so no effect can activate in response, including those of "Mucus Yolk" itself and "Gorz the Emissary of Darkness"

Battle damage as the name suggests comes from the damage step in the battle phase, technically it's like Blue-Eyes White Dragon destroying Dark Magician, the 500 damage is battle damage. There are two types of battle damage, one is done to players and the other is done to the monsters. In the scenario, Dark Magician, took 2500 battle damage that is why it is destroyed, the remaining 500 you took.