I am a long time Alaskan hunter and trapper.
When you use a #2 Victor leg hold - all deer can easily escape. The secret to these is to set your traps in place where the coyotes and deer are using for trails, but, lure your target animal off the trail and into the trap with food or scent. Deer eat grasses and have no interest in fried chicken pieces, or, gland lures from other cannines.
For snares it is a bit trickier. You need to discover the side trailes the coyotes and fox are using that deer are not. You want to place the snares in game trails through bruches, branches, deadfalls and thickets the coyotes are traveling though and the deer and other larger animals are going around.
It is important to always have a heavy coat and set of dykes with you. This way if you do get happen to get a deer in a snare or leg hold you can throw the jacket over the head - then very carefully snip the snare off.
Bottom line. Trap placement is key. You need to take the time to find places were the target animals you want to trap are the only species in a small area. If you take the time to bird hunt - the plucked feathers make both a great scent and visual attractant to coyote - that deer will have no interest in at all. Be creative!!
You can email me any trapping questions!