Replacing lost functionality is also acceptable for a cyborg, but Hawkings simply didn't replace any organic part of himself, he used external devices. Someone with cochlear implants or a total artificial heart totally should count as a cyborg though - they have integrated mechanical body parts (well, both are partly external I guess...)
But it doesn't really make sense that just because this version of a hearing implant doesn't give you better than normal hearing, having an identical device that did would be the difference between being a cyborg or not.