This is a really great discussion.

The original poster asked for suggestions for more experiments. I
suggest connecting the spectrum analyzer directly to the speaker,
don't use a mic. This will verify that this distortion is not caused
in the electronics. I doubt that it would be, but it would be nice to
verify.

The sidebands do look like FM. They also look like IM which I presume
you are calling AM. Note that both FM and AM produce symmetrical
sidebands. However when both AM and FM are present together, the
combination usually produces asymmetrical sidebands. This is because
the AM and FM sidebands have different phases above and below the
carrier. Since the sidebands in your experiment look pretty
symmetrical I would say that if AM and FM are present together, they
are not close to equal. If they were close to equal, the sidebands
would be very asymmetrical.

So how do we tell if the sidebands are FM or AM?

Suggestion. Send the signal through a 4 kHz BPF that is wide enough
to pass the sidebands but not wide enough to pass the 50 Hz or it's
harmonics. You should then be able to analyze this BP signal to see
if it is AM or FM. If it is AM, the envelope will vary at 50 Hz. If
it is FM, the envelope will be constant.

Here is the interesting part in my mind...
Regarding how this sounds, it really doesn't matter if the sidebands
are produced by AM or FM (doppler). The ear works like an FFT
analyzer. If the sideband structure looks the same, then it will
sound the same. Doppler is a non-linear process just like FM and IM
that produces new sidebands.

Mark