This is so interesting! I don't know for sure (I haven't played the game), but I have a couple guesses. :) 1. There might be other scenarios where you need to use more damage counters, and the instructions were including those scenarios too 2. Because there are multiple damage counter pieces that could be in play, and saying "damage counters" is easier/flows better than "the damage counter appropriate for the round" I would definitely choose "put a damage counter" in this situation, but "put damage counters" doesn't read as wrong to me -- because my next step would be to ask, "How many damage counters?" and "One!" would be a totally fine answer to that question.

It's probably referring to a generalization.

I guess it's because its a specific card the game uses damage counters instead of "one" or "a'.