by Piata
The rules make a clear distinction between damage and attacking. A monster is only considered attacking when it has at least one claw showing on the dice.i.e. If someone in Tokyo attacks you and you have reflective hide (which does damage to the attacker), they can't then abandon Tokyo because you're not attacking.
Going by that, anywhere it says "damage" means anytime you do damage. Anytime it says "attack" means anytime you do dice damage.