Language - in the human context - implies the ability to construct an utterance which has never been uttered before in the certain knowledge that it will be understood by all other users of that language. The important characteristic of human language is that it is infinitely generative.
My understanding, from my largely forgotten university psycholinguistics (four decades ago) is that people taught American Sign Language to chimps like Washoe mainly to see if it could be done. This was the time of Chomsky and the theory of deep structure and generative grammar. I think that researchers were interested in seeing whether these concepts were solely human. The idea that language is an abstract phenomenon separate - but related to - speech is an idea that has been around for about a hundred years. One of the problems of discussing non-primate behaviour in terms of warning signals and differential behaviour for differing threats can be explained without invoking language.
As for cognition being separate from language, I was watching my 20 month old granddaughter the other day. Her speech is - well - vestigial. She has about three words and uses them interchangeably. However, she picked up my iPad, activated it and was watching Peppa Pig within about 30 seconds. When her speech has improved she will be able to tell me how she did it.