> I haven't really used Delaunay Triangulation in this manner but by my basic understanding of the algorithm, doesn't it attempt to create an optimal triangulation, and therefore would tend towards outputting rather uniformly distributed internal points, rather than learning the distribution of the input?

Delaunay triangulation itself—not really, well, not in the way that would do much harm. We use it for simulations of mobile networks, e.g. analyses at the boundary between urban (where density of base stations is high) and rural (less dense) areas. If each triangle creates one additional point, regardless of whether you have a large triangle (rural) or a small one (urban), then denser areas will get more points. It won't lead to smoothly changing density between more and less dense areas, but then, it's an assumption you'd have to put in addition to your data, not infer from data themselves.

Judging from the visualization though, this algorithm seems though to have a stopping condition dependant on the size of a triangle, which breaks this reasoning.