Well, that video was fairly helpful, but it still didn't tell you how to hold the legs as you pick up the bird, and if you don't do this, your arms are going to get badly scratched by the flailing claws and this can lead to all sorts of pain and infection for you..
I always go up to the bird in the way shown on the video, and with the bird facing away from me, I put my left hand on her back to steady her, and slip my right hand under her tummy, palm upwards, and fingers pointing forwards towards her chest and spread apart. I slip the lower part of her legs between my fingers, (sort of from her knees down) with a finger or two fingers between the legs to cushion them, and a finger or thumb on either side of the legs. Then, when I have her legs firmly but gently, I lift the bird, supporting her weight on my right hand and along my forearm. My left hand remains over her wings to steady her, or I slip my left arm round the front of her chest and restrain her right wing. If she struggles, I just hold on to her legs and gently restrain her wings, pressing one wing against my chest and with my left arm round the front of her chest and over her wing. When the bird is calm, or with birds that are used to being lifted like this, it's easily possible to hold them with the right hand restraining the legs and the bird supported on your forearm, and no need to hold their wings at all. Some birds, especially large ones, are more comfortable with their heads tucked under your left armpit and their bottoms facing forward. This way of holding has the advantage that if the bird poos it will tend to fall away from you. When putting the bird down, I bend down to ground level, holding her, and let her put her feet on the ground and just walk away. This avoids the panic and possible injury of letting her fly down with her wings half-spread, scratching the holder as she does so.
You will find that, when your hens are in lay, they will be easy to catch as they will spread their wings and crouch down when you put a hand on their backs, in the expectation of the cockerel, and this makes it much easier to get hold of them.
It's very important to teach children the right way to pick up the chickens, not only so they won't get nasty scratches, but because children tend to hold the bird over the wings and hug her round the middle and squeeze when she struggles, which is very bad for her. If a hen is about to lay an egg and is squeezed in this way, it may even cause egg peritonitis if the egg breaks inside her, which is a painful and often terminal condition. I think it's good to pick up the chickens regularly, so you are confident that you can do it easily and calmly, and so that they are not alarmed when you do it and therefore you can examine them for parasites etc and check that all is well with them. But chickens really don't need to be cuddled - some people need to cuddle them, but my sympathies are with the chicken when i see pics of birds being treated like teddy bears!