For the ones with fences on the edges, is this strong enough to really stop or deter an infant from going off the mat?
This really depends on the strength and age of the infant. It is most likely that the toys will be better controlled rather than the infant, but the fence does present a barrier for the child to overcome. For a newborn, it should work well, but as the child ages and their physical capabilities become greater, the fence will not hold them back.