The rest of what you're saying makes sense, but only this sentence directly addresses the question of "what is a dominance hierarchy as opposed to other kinds", and while I understand the claim, I don't see a justification.

In the study of animal behavior, which is where the term originated as far as I know, we say that there is a dominance hierarchy if there is a pattern of deference regardless of how it is maintained. Because animals have a more limited set of options for maintaining the hierarchy, aggression (usually ritualized) often plays an important role. Partly, however, this is just because aggression is easy for humans to observe.

Pet rats typically establish a dominance hierarchy, but only in problematic cases is there much bullying per se. A relative of mine had pet rats and had trouble telling who was dominant, eventually realizing that the dominant rat would tend to sleep on top of the others (they like sleeping in a pile), and that the dominant rat would engage in territory surveillance and protective behavior more readily than the others. Unquestionably dominance when you looked closely (for example: the other rats wouldn't object when he ran off with the nicest bit of food, but he might go get it back if there was a clear disparity and they did; when they met another set of pet rats, he and the dominant rat in the other group puffed themselves up and were very haughty towards each other), but when the rat spent most of the last month of his (subordinate) brother's life keeping him company and grooming him as he fell victim to the diseases of old age, it's pretty hard to characterize dominance as maintained by fear and coercion. (Even then if you looked really carefully you could usually see the difference between dominant I-am-taking-care-of-you grooming and subordinate I-am-exalting-you grooming.) He was playing a role, and that included protection, care, and (probably) love, as well as maintaining the role with the right level of swagger for a rat to demonstrate that he belongs in that role.

Anyway, I don't necessarily object to using "dominance hierarchy" to include only hierarchies maintained by fear, coercion, and sometimes violence (then the rats do not qualify); but (1) I'm not sure this coheres with the broadest understanding, and (2) it then becomes incumbent to demonstrate that when you see a hierarchy these characteristics are present before you call it a dominance hierarchy.