How do I cull a chicken for the first time?
The way you can. Let me explain that. Your goal is to end the chicken's life as quickly and as sure as you can so it does not suffer. You also do not want to injure yourself, don't forget that part. We are all different with different experiences and sensibilities. What works for you might not work for me. With any of these methods there are details that can make a difference.
I use the stump and hatchet method. I grew up swinging a hammer, axe, hatchet, and other tools so I have confidence I can hit what I aim for. After you take the head off the chicken will flop around, that bothers some people. Happens with some other methods also. That's just reflex reactions, the head is gone so it can't be feeling anything, it is not suffering. But it really bothers some people to see that. If you are not comfortable using an axe or hatchet choose some other method, it would be easy to hurt yourself.
Many people use the broomstick method. If you don't pull hard enough the neck may not break. If you pull too hard the head can pop off. Again, if the head is gone the chicken is not suffering but some people are shocked when that happens.
Another method is the killing cone. A lot of variation on this one. You get a big cone, say a traffic control cone, invert it, and put the chicken in with the head sticking out. Then you can take a sharp knife and cut the throat. If you cut the arteries it bleeds out. Or instead of a knife use pruning shears or loppers to take the head off.
Some people shoot them. You have to be careful what is downrange, probably not feasible in town. And make sure you cleanly hit the target.
Some people use a CO2 box. Hook it up to an automobile exhaust. CO2 is heavier than air so make sure the box is sealed down low. I'd study the details on this one closely so you do it right.
Some people break the neck in their bare hands. I'm not exactly sure how that method works.
Of course there are other methods but that is enough.
As the chicken is suffering I think you are doing the right thing, hard as it is. One time I had to put a cat down that my wife really liked. I did not enjoy that at all and I would not let her watch. It was hard enough on her anyway just knowing.