Sound != music
Sorry, Johnny.
Say that sound is the physical phenomenon.
Sound is something that happens to you. (It can happen that you are making sound, but that’s just because the physical phenomenon is resulting from a willful act.)
Sound is your physical perception.
When that perception happens to you, you want to call it something. (You’re a person, and that’s what people do - call something “_________”.)
So you give it a name, and the name carries description, not just designation. (You don’t call it “phenomenon #47.”)
Maybe you call it noise.
Maybe you call it music.
In other words, sound is the name for something that happens to you. Music and noise are names for how you feel about what has happened to you.
What Cage did was not to make music == the set of all sound. Maybe what he did was make music and noise overlap. Or rather, he grew the term music so that it would completely overlap the term noise (without noise cancellation), as well as covering what it already covered.
That being said, never forget that Cage was absolutely a composer, and he definitely had a sense of what he considered noise (the undesired kind). For him, that arose when the means were suspect, perhaps more than the sounds themselves. (E.g. his stated views on improvisation, disregarding his practical relationship to it).