It is increasingly difficult to obtain horsehide and especially any that looks and performs well. I just recently sent back half of this spring's shipment...and half of that was replacement for what I send back in the fall.
Bellies and sides.....for cowhide the sides are fine if you cut from the rearmost areas, not up close to the neck where there are many fat wrinkles, hence not smooth leather (creases). Down near the belly it gets soft and stretchy. High near the back of the beast, closer to the rump is the best surface finish, thickness and duribility. Shoulders are usually prefered for belts, etc...because they don't stretch and are strong but I seldom find the surface quality I desire in shoulders for my strops.
On horsehide, the thickest, strongest leather is near the rump or butt. The sides and bellies wind up in expensive leather jackets and if it can be bought is usually skived to a very light weight, maybe 3 to 4 oz thickness.
I prefer latigo mainly for the draw and that it is less prone to cupping. The oils and waxes displace moisture, the most common cause of cupping. Veg tanned cowhide tends to cup more easily than the latigo processing. Bridle typically has a great feel but I feel it is more likely to cup over the long haul as it contains far less amounts of oils than latigo. My old Honey Brown was bridle leather but now, 5 or 6 years later the pieces hanging in my shop have no faired well while the latigo of the same age is still reasonably flat. each tannery has its own process though and the newer stuff may perform much better that what I used 5 years ago. I have not revisited that material though as i am content with the performance of horsehide and latigo so see no need to confuse the issue with too many choices.....high draw and low draw is enough for me.
My best advice for the DIY guy, go for a nice feeling surface, no fat wrinkles, no raised scars and don't worry as much about what beast it once covered. That is how i got started. I was already familar with different leathers and knew latigo was durable and stayed flat. It had a good surface so I tried it and I and my customers liked it. Since that point many makers have started using it as well.