I never know Miniature horses came in this many colors! I only expected a relatively a few colors and patterns in the breed, but I was wrong! Tobaino, overo, splash, perlino, palomino, roans, and appaloosas.
Minis are awesome. Not only do they have pretty much every colour, they also tend to have much stronger expression of most colours too. And in answer, the mini you posted is brown, not sooty bay
You can't mistake this counter-shading for something else. These are registered AMHA bloodline minis. I have seen a gene tested pure blood bay arab gelding with coloring very similar to this, and he was sooty.
Brown (and I'm not referring to Seal Brown) is a somewhat ambiguous term used to identify a number of horse coat colors but most often refers to a dark variation of Bay. We eventually will try to make changes to the calculator that will incorporate these variations. But for now, selecting Bay for the color brown will give you the most accurate calculations. However, colors such as liver chestnut or even smoky black are sometimes misidentified as brown. In which case Bay would obviously not be the correct selection.