Is glass an amorphous solid or supercooled liquid?
You are correct: it's not both.

Why? What you've heard is nonsense. Very old window panes are thicker towards one side because of how they were made at the time. It became common practise to put the thicker edge at the bottom, because it was less likely to exceed the acceptable stress on the thinner area.

Contemporary windows have been found with the thicker edge to one side, or even the top. Gravity on Earth has been consistently towards the core since at least the Middle Ages, so the "flowing glass" hypothesis is false.

Glass is an amorphous solid by definition (per Karl's comment) and is not a super-cooled liquid (per Prabhdeep Singh's comment).