I am assuming that this is because that sphere-like object is generated by subdividing and inflating an octahedron, rather than the "sphere" primitive found in a lot of 3d programs. Duckgoing for "octahedral sphere" brings up things like a tutorial on "creating better spheres by subdividing an octahedron".

Those meshes are made by subdividing an octahedron. Approximating an sphere like that has the nice property that the triangles in each subdivision level have the same surface area, and you can use the same underlying texture for all the levels since in the 2D projection the sub triangles are also regular subdivisions of their parents.

The second embedded YouTube video is probably a reasonable explanation. The full tree is replaced with eight (maybe nine, actually?) billboards.

N, S, E, W, NE, SE, SW, NW, and one looking down from above. That's nine.

Ok, I have no idea, either.

Those billboards are an alternative (the more traditional) impostor type. The octahedral impostors replace these billboards with a geodesic polyhedron of octahedral symmetry. Basically it subdivides the sides of an octahedron to approximate a sphere.

Not really. The octahedron defines the positions you pre-render the impostors from. The actual rendering uses a flat plane (like a regular billboard) but with a parallax shader and depth offset to make it look bent.

Yeah, I'm not sure what that's called but it ain't an octohedron[1].

The results look damn good though!

[1] https://en.wikipedia.org/wiki/Octahedron

'octahedral' could mean it has the same symmetries as an octahedron, not necessarily that it's an octahedron.