Damage is the number on the card the attacker is doing, or sometimes a number inside the attack text that includes the word "damage" but not "counters". Effects that stop Damage do not stop other attack effects (including damage counters being placed).
See for instance this compendium article:
Q: I've noticed that some cards say "Does 10 damage to the Defending Pokémon", whereas others say "Put 1 damage counter on the defending Pokémon". Is there any reason for the different wording? I've spoken to someone who says that you do not apply weakness or resistance for attacks where it specifies Damage Counters; is this the case?
A: There is a difference. As you mention, weakness and resistance are not applied when placing counters. Also, things like Defender and Metal Energy will not block Placing Damage Counters. It will not trigger things like Base Set Machamp's Strike Back. In the steps of resolving an Attack, "placing Damage Counters" counts as an Effect of the attack, it is not part of "doing damage".