What you describe is called an informal leader. But you are correct on everything else so far.
"Hierarchy" comes from "heira", which means "sacred rites". So it's "the rites which rule". In other words, "the process through which people are organized to do things"
To make it nice and confusing, "anarchism" has been applied to so many systems of beliefs over the years that it has no clear meaning on its own. To some people, it absolutely means "no structure whatsoever".
Etymology.com links:
hierarchy: https://www.etymonline.com/word/hierarchy anarchy: https://www.etymonline.com/word/anarchy