What would you say about a dog growling at you, communicating its displeasure at how close you are? If you back away, understanding what the dog intends to convey with its growl, does that make the dog’s growl language?
Is a honeybee secreting a pheromone to get the hive to swarm language?
If so, how is language meaningfully different from “communication”? And, is human communication with each other the same type of phenomenon as the cases you and I mentioned, or is there some sort of categorical difference there?
(Also, this definition isn’t classical - it’s quite modern. The tendency to conflate writing with language in cultures that have writing is as old as writing is, and disentangling the two is a relatively modern discovery.)