Youre fundamentally misunderstanding what a power structure is. It's not merely a group of individuals who are misogynistic (that is commiting acts of: violence against women, discriminating against women, subjugation women, and perpetuating hatred and prejudice against women) its a pervasive continuous problem across all levels of society and perpetuated by all functions of society. Misogyny exists so universally in our society that every single woman experiences it throughout their lives beginning as very young children. Our own parents teach us misogyny, our education system reinforces misogyny, our media shows us misogyny and so on. There's no woman who doesn't experience it, it affects all women as a class.
No such system exists that discriminates against men as a class.
I know, I get that, I'm asking about terminology. So what would you call a single person who hates women? Not the power structure, just that one person.
A misogynist, assuming you mean someone who is reinforcing or using the power structure of misogyny. To call it hatred is reductive, someone can be misogynistic and not think of themselves as a misogynist. They can have misogynistic opinions, commit misogynistic acts, or spread misogynistic misinformation without seeing themselves as someone who hates women.
Well that's a problem then, because you're using the same term to refer to two different but related things. Well, it becomes a problem when you consider misandry. Sure, there's no systemic oppression of men (except collateral damage from the patriarchy). But there are absolutely individuals and groups of individuals who hate or are dismissive of men. We need a word for that.
I think the popular definitions here are more useful than yours, because it prevents misunderstandings when someone says something like "misandry is a thing that exists". They're not saying it exists in a systemic, structural way. Just that there are individuals who feel like that.
I never said the word wasn't a real word, I'm saying that in the context of systemic discrimination against men there is no such system of misandry. That it is not true that body policing and control are not equal issues for men and for women. The thing I've been talking about the entire time.
Nouns can be applied in 2 different ways dependent on context, English works just fine that way. When we talk about misogyny in intersectional feminist discourse we are talking about the power structure of hatred and violence against women.