What causes the different types of tire skidding noises? Short: What mechanisms are relevant to the production of that noise and what are their parameters?

Up until a couple of years ago, I had only ever heard a quite, low, rubbing sound when the tires skidded on asphalt.

That was all on relatively standard sized tires (i.e. 28") and bicycles (i.e. upright non-folding or anything).

Now on my recumbent with 20" wheels when I brake sometimes I hear a loud screeching sound like on a car.

This happens when the wheel is skidding (i.e. speed difference of tire and ground somewhere between 0 (excluding) and forward speed), but not locked up (i.e. wheel not turning at all, speed difference of tire and ground equals forward speed). If it's locked up completely, then the sound is the quiet low rubbing noise I'm used to.

I haven't been able to pinpoint exactly what causes the different sounds. Maybe it's the asphalt type (new one seems to promote screeching?), pressure (lower=more?), wheel size (smaller=more?), wheel width (wider=more?) or braking technique (the noise mostly seems to appear when I get it just right and am only almost skidding).

What I can say is that I've been riding disk brakes for years and those specific disk brakes for some fewer years. I have readjusted them, changed pads, had annoying noises appear, made them go away again. This never seemed to change the screeching noise I described above.
The screeching that you describe is unlikely to be caused by the tyre skidding. It's more likely to be the contact between the brake pad and the braking surface causing the sound. 

On an alloy-wheeled rim-brake setup, this is the rubber pad rubbing on the rim. If it's squealing, then it could mean that the surface of the pads has picked up some shards of the brake track and is scraping them on the brake track. 

On a carbon-wheeled rim-brake setup, this could just be the standard sound. Some carbon wheels are just loud under braking and there's nothing to be done. 

On a disc-brake setup, this could indicate that the pads are contaminated, requiring replacement. If the pads are sintered, then like the carbon wheels with rim-brakes, there's only so much you can do. Sintered pads are just known to be a bit noisier. 

In any of these types, worn-out brake pads can also cause a screeching sound. If this is the cause, then you just need to replace the brake pads.