As the title will the putting damage counter attack such as Chandelure Ex "Cursed Drop" trigger Beedrill Allergic Shock? Since the wording is damaged by an attack, put damage counter look like different from normal attack damage. Thank you.
Yes, it'll be KO'd. In this particular case, damage is damage, no matter how it's applied.
The way I once heard it is that doing damage is like being punched, and placing damage counters is like God.
Humor aside, the origin of the 'damage' is different. For example, a punch to the face is going to feel a whole lot different than a cloud of smoke, right? The cloud of smoke would harm you but wouldn't be direct physical damage. A similar principle applies to Pokemon here.
Now I must go hang myself for making stupid analogies about a trading card game.