Humanely killing a chick that does not meet show standards, has deformities, and just won't live a very happy or productive life. Usually done by cutting the head off very quickly, but there are other easier ways.
while culling can mean killing it doesn't "just" mean killing. But also means to "Extract or pick out selectively". A person can "cull" birds from a breeding program but removing them for various reasons and not kill them. I intend to remove Nick from my breeding of Dells but I don't intend to kill him.
On the other hand it becomes necessary to "cull" or uthenize sick chicks or birds.
Yes, removing and killing are both correct. 99% of the time when referring to defects or deformities or attacks, the chicken/animal is put down(culled) humanely. When referring to breeding or thinning of flock it could be either selling or killing.