What's in a name?
Yes, the Black-backed is also three-toed, and in fact the Black-backed used to be officially named "Black-backed Three-toed Woodpecker", but that was deemed unnecessarily cumbersome and was later shortened. The other three-toed woodpecker used to be called the "Northern Three-toed Woodpecker", but when the "Three-toed" part of the Black-backed was dropped, the Northern Three-toed became known officially as just "Three-toed Woodpecker". (Why it wasn't called Northern Woodpecker was presumably because there are other northern woodpeckers, but then, as Joe points out, there are also other three-toed woodpeckers, so they should probably have come up with some entirely new name.) Then in 2003, the "Three-toed Woodpecker" was split from the Eurasian race into two separate species, whence our species became officially known as the American Three-toed Woodpecker. Confused yet?