I don't use an axe because I'm a clumsy cuss. I can't even swing a hammer straight and I've had more than my fair share of whacking myself, so I don't want to see what would happen with an axe!
I learned on pithing. If you're only doing one or two birds, we'd take the time to flip them over on their backs and hold them so they'd go into tonic immobility (also knowing as "trancing" or "hypnotizing" the chicken where they grow quiet and still and look like they've gone to sleep).
Then pithing was a snap. Open the beak and stick a small knife in while aiming between the eyes and give it a sharp stab and a quick turn once it's in. The bird shudders and goes limp after it's brain is destroyed. It never knows a thing.
Then you can slice the neck cleanly and leave it to bleed. Best part is, because the bird never has that adrenaline rush, after you scald it the feathers practically fall out on their own!
For doing more than three birds, I'd use the slicing method.