Ring the neck but slowly so as not to break the skin and twist the head off. If you do it slowly enough rotating the head and the body and extending the head out until the neck snaps, it won’t break the skin. Bird will still hang on a strap, not be a bloody mess, and still look presentable for a pic.
If you have a taxidermy bird then compress its chest until it "expires" no damage to the neck ruining the mount. Other than that I usually smack their head on something.
They make a little tool called the “finisher” or somthing like that. It’s got a 1.5” point and you push it in the back of their head and it kills them without messing up the feathers. I personally wring their neck or smack it against my boat, blind, gun stock, or whatever is around me.