I only knew there were two methods: wringing the neck and biting the skull. I've always just wrung the neck-- works fine for everything from woodcock to geese in my experience. I thought biting the skull was just something to get a rise out of the hunting partners. Suffocating a bird sounds brutal to me, even compared to crushing the skull.
Thoracic compression is a little bit more than just suffocating the animal. Properly done, it should also stop the heart by gripping it and compressing it directly.
Lloyd was looking for a humane method of finishing off a bird. The method recommended by the Nutty Professor essentially results in death by anoxia, or lack of oxygen to the brain. In a human, permanent brain damage from anoxia can result in 4 to 5 minutes, with death soon after. I don't know how long this process takes to kill a bird, but I'm quite sure it isn't nearly as fast as a quick snap of the neck. Personally, I'm not fond of the idea of biting a live birds head or skull, although it would be nearly instantaneous. But others might enjoy a mouthful of feathers.