I've killed birds two different ways: chopping off their head and cutting their jugular veins. Here's my opinion.
Cutting off their head = instant death. However, the birds flop a bunch. I used to drive two nails into a stump just wide enough to fit the necks but not the head. The stump was small enough diameter to allow the body to hang over the stump, while the neck was between the nails. I then placed a tub underneath the body. This allowed the neck to be extended by the weight of the chicken's body and provided a place for the body to drop was the neck/head was severed. I liked this way because both my hands were free to handle the axe. The problem I have doing this is that the birds flop so much that they frequently break wings, they end up covered in blood and are generally a mess.
Slicing their veins = near instant death. If I do both sides of the neck, the birds are dead within 5 secs (based on closing their eyes). If I slice only one side of the neck or do a crappy job, it takes 15 - 30 secs to die. I hand the birds upside down by a metal hook that I fashioned. The most important thing is that the hook hangs from a tree branch that far enough away from the tree trunk so that when the birds flap, which they do, they do not break their wings. I let the birds bleed onto the ground.
For me, both methods are equally humane but slicing the neck is my preferred method.