Pulling the neck is harder on you but much kinder to the bird. Suffocating her like that will make her suffer before she dies.
Hold her firmly against you, get a good grip of her head, between the head and beginnng of her neck, between your first and second fingers, look the other way and pull hard and twist. Do it firmly - it's much kinder. Or use the broom handle method described below.
I know it's horrid - we should never like killing other creatures, but it's up to us to get over our squeamishness so that our animals don't suffer more than they need to.