Both of these algorithms basically sum up the "signed area" of the polygons. This means that if you circle something twice, it'll count twice, and the sign depends on the direction of the winding.

The confusing part is that when the polygon is drawn, it uses the "non-zero winding rule" to determine which part to fill. So the filled parts of the polygons are all parts that contribute non-zero parts to the area (eg. positive, negative, two time positive etc.).

So the weird behaviour is that the physics simulation doesn't use the same rules as the visualisation!

So the nice thing about these algorithms is that it works for arbitrary complex shapes as long as the path has no self-intersections.

If you want to add support for self-intersecting paths, you need to decide how to deal with intersections. Presumably you'd want the physics to match the visualisation, ie. use the non-zero winding rule also for centroid and area calculations. To do that, you would first need to split the polygon into non-intersecting parts, and then calculate the area separately for each part, and then sum up the absolute values of the individual parts.

It was one of the grand challenges in the 00's, quietly solved by John Ratcliff. Then a few people solved it after him.

It's a fun algorithm. You basically slice and dice, like chopping a potato, then recombine small parts together based on a heuristic.