Thick sauce means it is heavy in texture, opposite of thin, not liquid. So it would not be like a broth (like chicken broth in chicken noodle soup which is very thin) it would be more like ranch dressing.
Pancake batter is thick and the opposite of thick would be thin, like how water is thin.
Rich sauce means that the flavor of the sauce is very high and flavorful or a deep flavor. It does not describe the consistency or the texture of the sauce. Rich is used to describe flavor, whereas "thick" is used to describe the look, texture, feel, and consistency of the sauce, NOT flavor.
Dark chocolate fudge sauce would be a rich tasting sauce and the opposite of rich would be a plain tasting sauce, like regular chocolate syrup.