"non-human animals are widely known to have language". I remember reading years ago about Koko the gorilla being taught sign language, but that Koko was incapable of seeing the sign for 'fire' without having a fear response. McCarthy didn't spell it out in the article more than to say:

"The simple understanding that one thing can be another thing is at the root of all things of our doing. From using colored pebbles for the trading of goats to art and language and on to using symbolic marks to represent pieces of the world too small to see."

I infer from this that a more sophisticated (or, in your words, specialized) definition of language that is of interest to McCarthy includes abstraction. I can talk about fire without arousing fear in a fellow human because they understand I'm not announcing the immediate presence of fire, I'm talking about it abstractly. As far as I know, no non-human animals are known to have the ability.