How are saturation, waveshaping and distortion different?

I was under the impression saturation and waveshaping were just types of distortion.

In an interview, Zomboy said that he doesn't use distortion, and saturation and waveshaping are key to his sound.

This had me a bit confused, cause I thought these things were distortion. Unless he meant this in the sense that he doesn't use distortion like, overdrive or something less transparent thats very obviously distortion.

But this did remind me, the sine and parabolic shapers in massive do produce a quite different sound than something like overdrive or a tube, it more changes the sound than plainly distorts it.