I don't think anyone has described this method for neck breaking, so I'll add it.
I hold the bird tucked under my left arm like a football, wings tucked in, feet haning out the back. left hand is across the chest, pressing the bird with my left arm firmly to my side.
right hand goes palm down (facing the floor) over the bird's head. with the thumb going under the jaw and the fingers closing around the head over the top.
the motion used to break the neck is sharp, firm, quick. you have to commit to the action, don't be tentative. to pop the neck, thrust the right hand down and out away from you, in line iwth the chicken's neck, sharply rotating your palm to face away from you as you do (thumb down, little finger up). you are both extending the neck and rotating the bird's beak up towards the sky. the idea is to break the neck at the point where it joins the skull.
this takes a sharp, committed movement, and a firm grip on the bird (both hands) but does not take a lot of strength. chickens are easier to kill than ducks, and this method may not work on larger birds like turkeys or geese, unless you've got a bit more power and longer arms.
mechanically what you are doing - how the neck is broken - is similar to the broomstick method, but the technique is slightly different.
so here's why I use this method...
as a kid my dad used an ax for killing ducks and geese. the ducks expire pretty quickly, but the first goose we did, although cleanly decapitated, got up and sprayed blood around quite a bit before properly expring. I know this is just nerve firing, but it freaked me out as a kid and it still makes me grit my teeth. so I prefer a bloodless kill, although I'm going to immediately decapitate to bleed the carcass out. I'm ok with blood, I just don't like it spraying around.
the other factor has to do with my heritage. I'm part native american, and I spent a year studying with an apache shaman... it changed and refined my ideas about my relationship to the world and everything in it.
please understand, this is my personal experience, and is not a criticism or commentary on anyone else's approach. and for those of you who are practical farmers - I totally get your position. I was pretty much farmer-practical before training with the shamman. now I'm farmer-practical, with an unexpectedly strong spiritual underlayment (at least it was unexpected when I discovered it mattered to me.)
for me, killing an animal is a spiritual act as well as a practical or veterinary one. when I kill an animal to eat it, or to end it's suffering, that's something I find to be a personal spiritual interaction between me and that animal, and I want to be as present as possible to that act. with our sheep or goats or any of our birds, I personally do as much of the killing as I have the skills to do effectively. if I can't do the job effectively myself, I am still present and participate where I can. with chickens and ducks, I can kill them with a neck snap in this hands-on way, and so I do.
one of the things I came out of the spiritual training with was an understanding of how unaware I had been of the life (and death) that suppored mine. the training gave me a profound appreciation for those lives (and deaths) and part of that appreciation, for me, is being participant not just while they live but when they die. I find shielding myself from that experience dimishes my respect for the animal, and that is something I no longer wish to do. the fact that it's uncomfortable for me is not relavant. the fact that I personally complete my responsibilities to the animal in as many ways as possible is.
it's hard to explain the experience of this process exactly, I just know that I didn't have this expectation or understanding before the training, processing livestock was purely practical, but that in some essential way the training changed me so that now that practical and humane aspects are part of a larger spiritual background.
it's probably odd. certainly some of my friends think so. but there it is.