Please dont call Chomsky into a debate about the word hierarchy, he has such bad takes when it comes to that. "Justified hierarchy" und such things...

Besides that, in my eyes we don't need unified definitions and theories. Its more important to make sure people unterstand each others words when going into a dicsussion/dialogue.

Critical Reading

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.

For me personally it means Systems of Domination or institutions of subordination.

One example might be that young boys are raised up with the idea that is their birthright to subordinate women when they become "the man of the house".

That dynamic does not exist in a vacuum. It is inextricably intertwined with capitalism, white supremacy, and more systems of domination, each which we should name and confront