Your little guy is adorable.. and he is simply bay. :) His sire passed on a black gene and his dam obviously can only pass on a red gene so your boy is heterozygous for black or Ee. He got either one or two copies of the agouti gene which is where the bay comes from. He didn't get a cream gene though or he'd be a buckskin (one cream gene) or perlino (two cream genes or double dilute)
Basic color genetics is not all that complicated if you break it down. From my understanding of it though there is no such thing as a "brown" horse, genetically speaking anyways. They may phenotypically be brown in color, but genotypically they are either black based or red based.
The only way to know for sure what color a particular horse is, is to have it tested.
A genetically brown horse has a different variety (real word is escaping me) of the agouti gene. There are only red & black base colors, so all horses are either red based or black based, everything else somehow modifies either the red or the black. So a true brown horse is genetically brown (black + brown agouti) just like a buckskin is genetically a buckskin (cream + agouti + black).