The idea that only coyotes eat the butt first isn't true. Vultures, bobcats, coyotes, etc... they all do that.
The only way to know for sure is to know your tracks.
Clear an area arround the carcass to where there is nothing but loose dirt (just loose enough for tracks to be left and readable). Take a broom (tree branch or weeds will work too) and dust the area so that the ground is clear of all tracks. check it about every 12 hours to see what tracks are left in the dirt then sweep again and check again if you are still unsure. If you can read tracks and tell the difference between bobcat and coyote tracks you should have no problem determining if you have a coyote at the carcass or not. It's fool proof and guarenteed to work.