Does the nature of the animal (domesticated vs. wild) affect the method of its demise for permissibility of consumption?

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The determining factor for permissibility is the state of the animal at the time of its demise, not its origin. If a domesticated animal becomes wild and cannot be slaughtered in the throat or jugular area, it is treated like a wild animal that can only be brought down by wounding, based on the comparison to wild animals. The ruling stating that if the head of a fallen animal is in water, it is forbidden, is because the water aids in the killing, resulting in a cause combining permissible and prohibited means, rendering it forbidden, analogous to a Muslim and a Magian both wounding an animal.