Why does aluminum foil on a pizza stone ruin the pizza?
To understand what's happening here, we should first have a look at what a pizza stone does.

A pizza stone is made from a semi- permeable material with a high thermal capacity, or, plainly put, can store heat and soak up humidity. This means it ensures constant heat at the bottom, plus it buffers the wetness of the dough, giving your pizza a crisp, yet fluffy bottom.

This is why your foil causes the problem. Where the stone can absorb water (and steam), the foil is watertight. Which, as you noticed, can mean a soggy bottom - the humidity from the dough and sauce has nowhere to go on the underside. 

Unlike the foil, the parchment is not completely watertight (especially not "steam-tight"), so the stone can still do its job.

If you have trouble with sticky pizza and like easier cleanup, parchment is the way to go. Alternatively, a generous dose of (coarse) flour, semolina or corn flour can help a lot. And don't worry if your stone gets a few stains, there is no need to scrub and clean it every time, just give it a quick wipe once it's dry again. From time to time, you can also burn off residue, if it bothers you.

A press toaster has a different working principle: Its plates are smooth and you are roasting your food only for a comparatively short time. And if you looked closely, you might have noticed a lot of steam either during toasting, but especially when opening it. So the foil has roughly the same properties as the toaster plates, hence you'll get the same result with or without it.