I'm looking for the name of this type of firewood. These are mostly made from limbs or small trees, about the size of a softball or smaller. Here is a link to a youtube video that shows two folks making small round firewood.
Cordwood is one term that well applies. It means wood cut into uniform lengths, usually for fuel though it can also apply when that sort of wood is used for construction as in cordwood houses.
The machine itself is a guillotine style wood cutter (not to be confused with guillotine splitter). The one in the video referred to in your question appears quite homemade but there do seem to be commercially produced (or at least commercial quality) guillotine wood cutters around, though they may be custom made by a manufacturer or independent fabricator.