Butterfly koi, also called longfin koi or dragon koi, are the one variety on this list that isn't a traditional Nishikigoi class at all, and purists are usually quick to point that out. Everything else here traces back through generations of Japanese breeding in Niigata. Butterfly koi were developed much more recently, and much further from Japan.
In the 1980s, American breeders crossed Japanese Nishikigoi with Indonesian longfin carp to try and create a koi with dramatically extended fins, pectorals, dorsal, and tail all flowing well beyond what a standard koi grows. The cross worked, and the result was a fish that kept the color and pattern varieties everyone already recognized, Kohaku coloring, Showa coloring, Ogon coloring, but wrapped in long, trailing fins that move like silk in the water. Because the finnage came from a non-Nishikigoi bloodline, butterfly koi aren't recognized in traditional Japanese koi shows and won't win against a standard-finned koi of the same pattern in a sanctioned competition.
That hasn't hurt their popularity outside the competitive show world. Butterfly koi are some of the most requested fish for backyard ponds specifically because of how dramatic they look in motion, and the long fins make even a young, inexpensive fish look striking in a way a same-age standard koi usually doesn't. They're also generally just as hardy as standard koi, since the health traits came from otherwise normal Nishikigoi stock.
For keepers who care about show lineage and strict variety standards, butterfly koi will always be a separate category, sometimes even a controversial one. For everyone else building a pond to enjoy rather than to compete, they're often the fish that gets the most comments from anyone standing at the water's edge.