Maybe they called it "face" and not "edge" because an edge is normally understood to be what's between two vertices (of a graph; so an edge has two vertices, beginning and end), while here "face" is what's between two corners of a given triangle (so a face can contain more than two vertices, and so multiple edges).

See the bottom "face" of the top centre triangle in the 4 examples.

Yeah, that's exactly it! "Face" is the term used in the original paper, so that's what I was using. I've updated the page to make the distinction clearer. Thanks!