When we say “pain” do we mean the firing of the nerves, or do we mean awareness of it?

I'd say awareness of it, especially considering AFAIK local anesthesia (as in, fully awake during) doesn't stop the nerves from firing, just from the signals getting to the brain, and I'd say that's blocking the pain.^

Also there's an interesting point in there: (this is not a rebuttal, just an addition)

Doctors used to assert that babies didn’t feel pain, because 1) they couldn’t tell us about it, and 2) they didn’t remember it later.

The reason doctors nowadays use anesthesia, (besides it being squick-y not to) is because it was discovered that, (though they may not consciously remember it) it can manifest in PTSD and other trauma related disorders later in life.^

In addition, it was found that there is a heightened level of stress hormones during surgery on infants, which were absent in surgery done with anesthesia.^ (These stress hormones also increased the risk in post-operational complications, due to the hormones acting to break down carbohydrates and other fats in the infant's bodies)