A koi betta is simply a commercial label (strain name/nickname) for a specific coloration of a marble betta; usually a cellophane based body with red and black spots of color. Like a koi fish!

So, what's a marble betta?

Marble is a name for a transposon (jumping gene): a DNA sequence that can change its position within the genome. As a result, marble bettas often have splotches of pigment (or areas without pigment) all over their body and fins. Because the marble gene is a transposon, its color is unstable, and often marble bettas will develop more or less color throughout their entire lives; a result of the gene "switching" pigment on or off. A transposon cannot transform into a normal gene, so there are no marbles that are guaranteed to hold their color pattern throughout their lives. It can affect all pigments on the fish, making for a literal rainbow of possibilities.

A koi betta is a marble betta! However, not all marble bettas are koi. There are many commercial labels for popular color mixes of marble bettas, pictured below.
(Click picture for original source.)

Fancy
Marble with 3 or more colors.
*Koi's are "Fancy."

Galaxy
Marble with partial iridescent scaling.

Candy
Marble with black, red, & blue or yellow.

Often breeders will combine the commercial labels together, such as "Fancy Koi Galaxy" which just means it is a betta with red, black, and iridescent marbling.

Tiger
Yellow bodied fish with black or blue marbling.

Nemo
Marble with orange, red, and black.

Samurai
Marble with partial dragon scaling.

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Iridescent

Black

Red

Cellophane

Marble is not a color, rather it's a gene that affects the color layer it resides in.
It affects the top 3 layers of pigment individually.

Let's look at this candy marble:
We can see he is showing these colors: ⬤Bl⬤bm⬤R
We also know he carries the marble gene: Mb
The Mb in his iridescent layer has removed some ⬤Bl pigment, exposing the black layer.
The Mb in his black layer has removed some ⬤bm pigment, exposing the red layer.
The Mb in his red layer has removed some ⬤R pigment, exposing the flesh tone in the cellophane layer.

We currently do not know if a separate marble gene exists for each color layer. It's hypothesized because the first marble bettas were only black and white. In these bettas, the red layer was skipped entirely, which suggests that multiple different genes may be at play.

We cannot tell, without testing, what genes our bettas have. We can only observe some of the traits they express (phenotype.) Due to the nature of transposons, a betta may appear to be completely solid colored and still be a marble. This makes it difficult to tell if the fish is just a carrier of the marble gene, actually a marble, or just a solid colored fish.