In particular, I expressed doubt about the definitions of lune and lens given at MathWorld: <http://mathworld.wolfram.com/Lune.html> and <http://mathworld.wolfram.com/Lens.html>. My guess is that the distinction based on whether the radii are equal or not is solely due to Eric Weisstein. OTOH, I am also slightly uneasy with my suggestion there that the distinction between lune and lens should be one of convexity. After all, an optical lens need not be convex in section and, when the Moon is gibbous, the lighted part that we see from Earth is convex. Partially due to my uncertainty, I have not yet written to Eric to try to get his entries for lune and lens corrected. If anyone can shed light on what is "correct", historically or otherwise, I would appreciate it!
Given two circular disks, A and B, having a nonempty intersection and such that neither is entirely contained in the other, three regions are formed. (Think of a Venn diagram.) One of the regions, A /\ B, is convex; I suggest that "lens" be used to name that kind of shape. Neither of the other two regions, A-B and B-A, is convex; I suggest that "lune" be used to name that kind of shape.
I also suggest that a circular disk itself should be considered a degenerate case of both the lune and the lens.