There always seem to be so many disagreements between colors and shades in the horse world. Regardless of whether we think sorrel and chestnut are the same or not, at least in the APHA world, they are recognized as different colors with regard to registration. You can see the APHA examples at
APHA.Com - Coat Colors
Our mare, Cinnamon, is registered as a sorrel, and our mare, Lady, is registered as a chestnut.