Equilove, even the horse you posted is a brown.
Brown has been found on the genetic level. It is a form of agouti along with bay and wild bay. Brown horses are lighter around the muzzle and in the elbow and stifle area. They can vary in body shade just like any other color. Brown horses can and do have black points like any other horse with the agouti gene.
ETA a registry is the absolute last place I would turn to for a color definition. 99.9% of them are so far behind on the color genetics it isn't even funny.