Outside of spreading the definition that I've given, I don't know what else you'd need or want to do. I'd like to think that this comment thread doesn't really qualify as an 'anarchist theory rabbithole', so if the explanation I gave worked here then perhaps it can work elsewhere. I don't think that this definition is likely to be controversial.

I'm not sure what you are asking for. I'm just saying that it would be good for people to understand hierarchy in terms of coercion, generally, so trying to establish that definition as common knowledge would be a good idea.

I think it would be a very good idea to make the link between the notion of 'hierarchy' as anarchists like to think of it and coercion a part of common knowledge, both inside and outside of the anarchist community. I think everybody being on the same page with terminology would help clear up a lot of miscommunication about anarchism.

I don't think that this is the primary problem that we're facing, though. I think that part of the reason that it is difficult to pin down a definition of 'hierarchy' that everybody agrees upon is because there are some 'anarchists' that don't actually want anarchism, but instead want a lack of personal accountability - in other words, the freedom to do whatever they want to whomever they want without consequences. If you frame this in terms of 'personal freedom for everybody', it sort of sounds like anarchism, but because it emphasizes positive freedoms to the point of discarding negative freedoms almost entirely, it actually ends up being a self-contradictory position where bullies have power because they're willing to penalize their victims into submission, and there are no collectively-enforced consequences for engaging in such oppressive behavior.

I think that the #1 problem that we need to solve is the issue of these individuals generally being an accepted part of our group (or society at large, even). Anarchism isn't actually a magic bullet for oppression, because even an anarchist society would eventually be corrupted into an oppressive one if bullies like this are allowed to persist and manipulate people into following them and their disordered ideology. To actually create a truly 'good' society, we need to learn how to reliably recognize these bullies and keep them out of our spaces. Anarchism helps enormously, because collective power is much harder to subvert than hierarchical power, but it isn't a complete solution on its own.

That said, being able to recognize coercion and manipulation in all of its various forms would help a lot with that goal, and so the goals of establishing such a common terminology and also teaching people how to recognize bullies in all of their various forms are synergistic.

Hierarchy happens when some people are ‘above’ others, or, in other words, can make decisions for other people and enforce them without their consent.

I intended for this to basically be the definition, but I can try to rephrase to make it clearer.

'Hierarchy' is any social structure or relationship where someone has coercive power over another person, and where that coercive power is a normalized part of the structure or relationship. This normalization could be a social contract or the result of patterns of abusive behavior, it doesn't really matter exactly how it happens. It's just important to distinguish between coercion that happens because someone is breaking the rules and coercion that happens even when nobody is breaking any rules. It is the latter that forms a power dynamic between individuals or groups, and it is these power dynamics that hierarchies are made of.

I hope that clears things up.

It is tricky to do this in a way that allows for complex coordination at scale while avoiding the oppressive properties of systems which we would all recognize as undesirable. The concept that is required in order to properly navigate this is 'coercive control'.

Hierarchy happens when some people are 'above' others, or, in other words, can make decisions for other people and enforce them without their consent. This is coercive control. This is the situation that we want to avoid.

At the same time, the collective does have to be able to enforce rules, so some degree of coercion must be allowed in order to avoid a sort of paradox-of-tolerance situation. To resolve this apparent contradiction, we introduce the notion of a 'social contract'. To be a part of an anarchist organization would require that a person agree to a social contract. As long as the person upholds the social contract, they cannot otherwise be coerced, as that would be a violation of anarchist principles, and could result in oppressive behavior. (Note: For this to be self-consistent, coercion has to be outlawed as part of said social contract.) If a person breaks the social contract, then they are also no longer protected by it, and can be coerced by the group to leave or such.

The social contract bit is sort of an aside to the original question, but I think people get confused by what is and isn't hierarchy because they understand the first part in some way, even if they don't have the words for it, but then they don't know how to solve the obvious problem of enforcing rules so that people's freedoms can actually be protected in practice.

Centralization doesn't necessarily imply hierarchy, as, given an undirected acyclic graph, you can pick any node to be the 'root' and end up with a valid 'tree' - the structure that most people would visualize when they hear the word 'hierarchy'. Of course, we would prefer to not pick any node as the root, as the hierarchical structure implied by such a distinction shouldn't be necessary for the collective to reap the benefits of such centralization. There is a whole discussion that could be had about how to actually implement centralization without falling into various traps, but that isn't really what the question in the OP is about. I simply bring it up because some people confuse centralization for hierarchy, and end up shooting themselves in the foot for doing so.