Are Long Fin Koi actually koi?

 Short answer, yes.  However there are those who will argue the point.  I guess it depends on your definition of "koi".  Some will say that only those that conform to the conformation standards set forth in the judging guidelines are koi and all others are not.  I suppose that is a valid argument.  And by that definition longfin koi and butterfly koi do not fit those standards and are therefore not koi.  But really lets look at this a bit closer.  Longfin koi were developed by an Indonesian river carp that had long flowing fins.  The river carp was genetically close enough to breed with standard koi and produce fertile off spring.  So right there is proof that they are very close genetically.  But to take it further, koi and longfin koi have been bred and cross bred hundreds of times over in order to achieve the same patterns and colors as regular koi.  Also, longfin koi are hardier then standard koi (because they are not as many generations removed from that wild longfin carp) and many Japanese breeders have crossed longfin koi with their existing parent stock in order to make their bloodlines more resistant to parasites and infections.  So even a koi that appears to be a perfect standard fin koi may have some longfin grandparents. 

So to answer the question, genetically speaking longfin koi are practically indecipherable from standard fin koi.  But they are different looking and therefore considered a different type of koi.