Well the second one also looks dark brown to me, I see brown, not red. I'm sure some people would argue that she has enough brown on her belly and flanks to be bay. But I tend to stick with bay being more a reddish color and less a brown color.
In general, in the US, people are loathe to call a horse 'brown' or 'dark brown'. In Europe, I think there is just less emotion around color, so you hear people using the term brown for these colors.
But here, oboy. Let it be called anything, faded black, smoky black, ANYTHING but just plain old BROWN, which sounds so boring. We love to cook up lots of names for colors.