It's an ontological argument. OP is creating a categorical distinction where "sound" is the cognitive process by which pressure waves are perceived, eg as information. I think it's a fairly common distinction to make, but it is also kind of unsatisfying is the sense that it feels a bit like linguistic nihilism.
But neither tinnitus or cochlear implants have any vibration associated. If they are sounds then sounds are more than just vibrations. At the same time, not all vibrations are sounds.
The argument is that sound is part of our internal processing of sensations. If there is no brain to perceive it, is it a sound, or just a vibration in the air?
Yes and no. It all depends in what field you're describing sound. In physics, a tree that fell in the forest most definitely made a sound. In psychology, it doesn't.
In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the reception of such waves and their perception by the brain.