Ok, I never tried to program one so I can't tell if we're discussing different characteristics, but based on the register set and brief documentation I read, it seems to have different envelopes and volume settings alright, but not so sure about waveforms or pulse width. Even the POKEY is said to have variable duty cycle on the square wave, which I think is baked into the noise register settings so you can get a different pulse width or white noise, but yet again that is something I've only read about.
Waveform to me is a timbre, like the SID that has triangle /\/\/\, sawtooth |\|\|\, variable pulse width |_|___|__, combinations of those and white noise. Envelope is a function of how volume changes over time, and usually contains an attack phase (fade in), a decay, a sustain level and a release level. Some more advanced synthesizers may have multiple attack, decay and sustain levels for the same note, and the envelopes on the AY seem to repeat an ADSR cycle multiple times without retriggering the note manually.
Btw, as part of the so-called Viznut Waveforms, a couple of years ago it was realized that you can synchronize the internal timers in the VIC-I chip too, and get kind of different pulse widths on the VIC-20. In the old days, people would notice that suddenly the sound got a different timbre and then back again without knowing exactly why or how it could be used, but this behavior was tamed and I think even VICE emulates it reasonably these days. There may be internal timers in other sound chips too that you can sync and cause different pulse widths, I don't know.
Ok, I thought it had square waves and several sawtooth waves (|\, /|, /\) but that is probably some other chip I looked at.
The envelope can repeat or be "one shot" if I remember right.
The example I posted is a modern creation which takes advantage of newer trackers I'm sure. Older stuff doesn't sound as good. There are literally hours of older AY tunes on youtube you can check out.