I know I am out on a limb here, but to me, a flat bow is what I think of when I see about 90 percent of all longbows on the market. They have wide, thin limbs that are basically flat.
They are not as wide and thin as a recurve, but they are still wide and thin.
When I look at one of my Hill bows, I do not see a flat bow. The limbs on my bows are nearly as deep as they are wide.
In my simple, pea-sized Idaho brain, there are flat bows, and there are deep-cored bows. Nothing wrong with either, but they are different.
I recommend taking my thoughts on this with a grain of salt though, as I also think Hill bows look good with bow quivers. My thinking on such matters is what some would call "Abby-Normal."