The green loop on the front page is the signal. In against out. If a saturator were memoryless it would draw a line, not a loop, because every input value would map to exactly one output value and back again.
It opens up because the high-pass sits after the saturator. The filter has state, so the way out is not the way in, and the curve you get on the way up is not the curve you get on the way down. That lag is hysteresis.
Why model it rather than fake it
You can approximate the sound with a static curve and a bit of filtering either side. It will measure close and it will not feel right, because the thing that makes saturation musical is exactly the part a static curve throws away: the output depends on where the signal has been, not only where it is now.
So we model it. The behaviour falls out rather than being drawn on.
Try it
Drag the curve. Left and right moves the high-pass step, up and down changes the drive. Everything in the telemetry panel is the real thing, computed per frame.