The three main horse colors are Red (variably called Chestnut or Sorrel), Agouti (Type 1 Bay or Type 2 Seal Brown) and Black.
Then you have modifiers like Dun, Cream, Champagne, Pearl, Silver, Roan
Patterns like Frame, Tobiano, Sabino, Rabicano, Splash White
Leopard complex (appaloosa with or without patterns)
Greying gene
Dominant White
Some colors have special names.
Black Dun is Grullo.
Cream Black is Smoky Black
Cream chestnut is Palomino (and a dun Palomino is a Dunalino).
Double cream chestnut is Cremello.
Cream Bay is Buckskin (and a dun Buckskin is a Dunskin).
Double cream bay is Perlino.
Black roan is called "Blue Roan"
Black roan with Silver is called "Silver Roan"
Chestnut Champagne is Golden
Bay Champagne is Sable
Black Champagne is Classic
Apricot is a special color that requires the pearl and cream gene both.
Grey foals are always their base color and turn lighter grey throughout their life (they change at age 3 on the game).
Then there are variations of colors.
Buttermilk Buckskin (more "golden" color)
Chocolate Palomino (darker color)
Flaxen Chestnut (blond mane)
Liver Chestnut (darker red)
Blood Bay (darker red body)
Sooty (darker hairs usually on the back)
Pangere (lighter hairs usually on the stomach)
And there are colors/markings that are not defined by genetics here (not directly inheritable on the game, though they may have some genetic basis in the real world)
Birdcatcher spots (white spots)
Bend-Or spots (black spots)
Chimera (when twin embryos fuse into one foal)
Brindle (which may be a variation of chimerism).
I'm sure I'm missing some, but you can see that all the genes in the list past E and A are just modifiers on the basic colors.
Heavy Snow, Limited Visibility
