Jack is a hokahu. At least I think so - I don't see any flecks of black in the photos.
A kohaku is a two color white fish. So white with red steps. The steps should be bold and not freckles, unless you have a special kind that looks like flower petals. The effect should be pleasing, but need not be well balanced. Some of the more energetic pattern types can be very desirable. Like the "inazuma" or lightning bolt pattern.
A single color koi could be called an ogon. In the USA because we do not value single color koi, we call them bait.
(that's a joke).
Some ogons have special names.
A black koi is called karasu (translates roughly to "crow").
A shiny white ogon is a purachina.
A shiny yellow ogon is a yamabuki.
But a matte yellow ogon is a kigoi.
In general, the extremely shiny koi are called harawaki - but it would not be correct to say a kigoi harawaki - that would be a yamabuki.
Many of the single color koi also have a reticulation (a darker or contrasting color just on th edges of the scales). Then you add matsuba to the name.
Another way to say single colored is muji. So aka muji is a solid red fish. The term does not imply quality, tho.
So aka muji is a single colored red fish that we would term bait. While beni goi means a deep red, beautiful single color red fish.
