Alright, I'm at an actual computer now so I'm to go through bit by bit and you can tell me where my apparent misunderstanding is.

Your sources do a lot of dancing to avoid defining their principle ideas, and mutual understanding of concepts is integral to constructive discourse, so I'm going to do my best here: Anarchy opposes coercion, authority, and hierarchy, particularly that which comes from a state.

It’s a lack of hierarchy, but that doesn’t mean it doesn’t have government, rules, and protections.

Right away we have problems. The concept of free-association does mean there are no rules or protections. Not real ones anyway. Rules and regulations require an enforcement authority or else they are merely suggestions. You are free to make a rule and someone else is free to ignore it. What gives you the right to enforce your rule?

If something does grant you the right or ability then that thing, whatever it is, is a hierarchy of power.

Government is heirarchy [sic]

It is not. It can be, but it is not a fundamental aspect of it.

Government is a tricky thing to nail down because it covers a wide range of scales and intents. At its most basic the idea of governance is the codification and/or centralization of rules and processes. This can be the bylaws of a small cooperative or the many branches and layers of a nation. The single common thread is that the body exists to do something in lieu of or at the behest of a greater population, it is an alternative to direct democracy. This means that the government body has the authority, granted or taken, to represent its constituents.

Compare this with a think-tank, where the group exists to make recommendations but has no power to create policy or enforce on their own. This is not a government.

People can cooperate without the need for a hierarchy.

Yes, this is what I mean when I said "Anarchy [is] only possible if everyone engages in good faith.

They can agree that some actions are bad and to punish people without an elite doing so.

So they grant themselves the right to enforce their will on others, and you say this isn't "authority"? This not a hierarchy of power, an organized group coercing behavior through violence? What of the consent of the governed?

I like the ideals, and I support them inasmuch as this kind of cooperative and stateless utopia is the theoretical goal of classic Marxist Communism, but this freedom requires a much higher level of trust and knowledge than I think humanity is capable of. Opposing all forms of authority now, when we're facing the existential threats of climate change and broad resource mismanagement, is a mistake. Now we need people with the means to reverse course, with the power to enforce policy, and with the speed and focus to work before it's too late.