A sine wave is a single harmonic.
A saw wave is multiple harmonics (both odd and even).
Each harmonic in a saw wave is a sine wave; thus, you could think of a saw wave as just being many sine waves playing at various amplitudes and frequencies.
A square wave is multiple odd harmonics.
Depending on the saturation plugin, you’re adding in odd and even harmonics to the source sound. This is what we refer to as ‘saturation’, the addition of harmonics.
A distortion that evens out odd harmonics provides a saw wave type sound.
Tape saturation = squares the wave, a distortion type that sounds more like a square wave.
Joe tends to stay away from complex waveforms and builds bass from sine waves, using distortion to add in harmonics.
Wave-shaping = reduces the fundamental frequency and emphases upper harmonics.
The distortion types in Ableton’s Saturator are all square wave types, other than the Sinefold.