I cut the head off a chicken with a hatchet. Or a machete. Or an ax.
I've been doing it that way for years, following the example of my parents and grandparents.
I've eaten plenty of home-processed chicken, and plenty of chicken from the store. I haven't noticed any problems.
I've seen other people recommend various scissors-style cutter, or cut the head entirely off with a sharp knife.
Since I already knew a headless chicken will bleed out just fine, I've never paid much attention to the people saying otherwise. I can see how it would be confusing to someone who is trying to learn about the subject for the first time.