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Black vs. Brown and Bay Horse color

If a Horse is either homozygous or heterozygous black, but has one or two copies of Agouti (Bay), then the horse is brown (Bay) not black. 
If a horse is Homozygous black or heterozygous black, but has NO Agouti (Bay), then it is black. 
       Here is what Animal Genetics says to which backs this up:
  
 

Equine Color Coat Testing-

Base Color:

Every horse has a base color, which can be black, bay, or red. This is controlled by the Extension (Red/Black Factor) and Agouti genes. The Extension gene controls the production of black or red pigment throughout the coat. The allele for black color (E) is dominant over the red allele (e), so a horse only needs one copy of the black allele to appear black-based.
 
The Agouti gene can then modify black pigment by pushing it the points of the horse, creating a bay. The Agouti gene is dominant, so a black pigmented horse only needs one copy of the Agouti gene (A) to appear bay. Agouti does not have any effect on red pigment.
 
There may be some variation in the intensity of the base colors, for example, dark bays compared to light bays or liver chestnuts to sorrels. This could be caused by a variation in the expression of the genes or interaction of other genetic factors.

Horse Base Coat

Red/Black Factor

Agouti

Solid Black

EE or Ee

aa

Bay or Brown

EE or Ee

AA or Aa

Red (Chestnut/Sorrel)

ee

AA, Aa, or aa

 

 

     The difference between bay and brown is a modifier to the Agouti Gene. "At" is the Agouti gene reference for brown. Currently the other modifiers to the Agouti gene all equal bay, but At is brown. The Bay gene is dominant. So a horse with AtA will be bay yet carry the brown gene and could throw brown or bay babies, but a horse with AtAt or Ata will be brown and can only throw brown.

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