That did the trick for me, too. I wonder exactly what DiscretizeRegion is doing to the RegionPlot object. To sum up, I'm guessing that native objects like Disk and Rectangle and simple Polygons have DiscretizeRegion somehow built in, but a Polygon construction with more complex data needs to be discretized. I include the revised figure.

The requested piece of code is the code for g2 which produced figure [b]. Does my code produce in your system a figure [b] with a white crescent with a diagonal line through it as in the triangle in figure [c]? If so, then I would conclude that the problem is the MMA 10 rather than with the code. I have isolated the code for figure [b].

sz = 1.5; (* rect corner dimension *)

c = 0.551915024494; (* Bezier constant for circle *)

bz = {{0, 1}, {c, 1}, {1, c}, {1, 0}};(* Bezier control points *)

bzF = BezierFunction@bz;

bzFPts = Table[bzF[t], {t, 0, 1, .1}];

crescentPts = AppendTo[bzFPts, bzFPts[[1]]];

figure = Line[{{-1, -1}, {1, 1}}];

rectanglePts = {{-sz, -sz}, {-sz, sz}, {sz, 
    sz}, {sz, -sz}, {-sz, -sz}};

(* error message generated here *)

regionDiffCrescent = First@RegionPlot@RegionDifference[
     Polygon@rectanglePts,
     Polygon@crescentPts
     ] ;

(* remove the default directives from new region *)

maskCrescent = DeleteCases[regionDiffCrescent, _Directive, -1]; 

g2 = Graphics[{figure, FaceForm@LightGray, maskCrescent, 
   Text["[b]", {-.75, .75}]}, ImageSize -> {100, 100}] 

The new feature is good to know about, but I'm still using version 10. Can one conclude that DifferenceRegion also fails with these Bezier data in version 12?