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).
The catch-all term for these is"plane." Circular plane, square plane, rectangular plane etc.
A disk is defined as a plane bounded by a circle. Searching for the phrase "plane bounded by a square" does not yield a single word for which this is the definition.