Homonyms are two words that are spelled the same and sound the same but have different meanings. A definition is a statement of the exact meaning of a word, especially in a dictionary.
Cool as in cold must be a different word than cool as in calm. They're homonyms. If homonyms are two words that are spelled the same, spelling must be an arbitrary attribute of a word.
So, if I go find "cool" in the dictionary and see there are several different words with different meanings listed, what is "cool"? It doesn't refer to a specific word without context. Is it a phonic phenomena? What do I call "cool" in reference to the set of possible words it could be?