I agree that in a single one of the frames (the second frame shown in the first animated GIF above) from "The Naked Time," it looks convex. But the frames in which it looks the other way to me are outnumbered by an order of magnitude—namely, all the others.
I suppose another possibility is that it's a thin clear plano-convex lend but flipped around, flat face outward and curved face inward towards a matte black disc through with a light mounted in a through-hole. I'm going to dig up such a lens and experiment with that hypothesis today. I don't have a good sense of what kind of light play it would create, but it's possible it could create the illusion of this circular ridge.
This would be an example of that phenomenon
This is a plano-convex lens. This geometry seems to create a harsh edge rather than the appearance of a smooth circular ridge like we're seeing sometimes, so I'm thinking maybe bi-convex.
I didn't have lenses of the right kind on hand to experiment with, but I've got some coming.