When talking about circles, a "circle" refers merely to a line in the shape of a circle, whereas "disk" (or "disc") refers to a "filled circle". Is there a similar distinction for squares or rectangles?
"disc" means "thin, flat, circular object" and is mostly applied to 3D objects. "Circle" is for 2D-ish circular things.
In math, "disc" has its own meaning; the meaning you mentioned, as well as the normal meaning.
Names of shapes cover both definitions you provide ("a line in the shape of X" and "filled X"), except circle (and in non-rigorous talk, they get conflated too).