My description of hum versus buzz was a non-technical way of dealing with sine waves versus other wave shapes. If you try to make the sound mmmmmmmmm and keep your mouth closed, that's hum. If you try to make the sound eeeeeeeeeee with your cheeks pulled back, that's more like buzz.

I think of hum as being a sine wave -- there's 60 Hz and nothing [much] else. If you rectify 60 Hz, you get 120 Hz, but it's no longer a sine wave. Instead it's two half-sine waves in the time of one 60 Hz sine wave. The sharp transition of voltage change at zero volts adds LOADS of overtones to the waveform, and thus it can't be hum; it's now buzz.

You can also have 60 Hz buzz if you distort the waveform. The waveform of voltage across a neon bulb will be 60 Hz buzz; 60 Hz audio run through an amp until the amp clips gives you buzz, too.

The original problem sounds impossible, by the way. I'm reminded of the guy who had small speakers for a few years, then bought a subwoofer. He called a bit later to complained that it hummed.  I had him try some things while he was on the phone and it hummed, no matter what, including being unplugged from the wall and from the system.  I went to his house and determined that since he had bought the subwoofer he was carefully listening for low frequency sounds and he was now noticing, all the time, the refrigerator in the kitchen on the other side of the wall from his small speakers.  He was focused on low frequency sounds and suddenly heard something he had ignored for at least five years.  We have to be careful what we think we hear.

When I explained it to him, it was sort of a "what are you going to believe, me or your own two perfectly good ears?" moment.