the guy with the bigger stick making the rules

aka a monarchy

That's why anarchy isn't stable, it's a state between governments but eventually some kind of rule will emerge, and you're correct in that "bigger stick" is likely to be the first.

Anarchy in our reality is basically just warlordism. But when people talk about being an anarchist, they usually mean something like anarcho-syndicalism or anarcho-capitalism.

I’m not an anarchist so I’m not going to do well explaining it but usually, the “anarchy” part is a step towards a larger transformational goal. No one is an “anarchist” in the sense of wanting society to collapse. The “anarcho-“ part is them saying a layer of power (like the nation state, for instance) is unnecessary.

We have a better word for that. Dictatorship.

The short of it is "power to the people".