i do not agree with this vid. I freeze my older ones and have raised meat chickens. The easiest and most human way is just grab them by the legs and put there head on an old log. They will stretch there necks over the log. Have a sharp Axe and one swing its all over. Think about it. Would I rather have my throat slit or in one easy motion have my head cut off. I'll take the second.
Unless you are managing to perfectly sever the brain stem from the brain every time then the two deaths are scientifically the same, as conciousness will last in both cases for a few seconds until the brain is deprived of oxygen. A beheading at the throat is a quick death but not an instant one.
I think picking the method you are most comfortable with is important. I pick the way that I feel personally I have the least chance of messing up based on my own tools and skills. That is my biggest fear in the process. People who are used to another method are likely to have better success doing that. There are a variety of perfectly fine ways to go about it.