On the internet, it is common to call a guy a misogynist, but what is the exact meaning of misogynist? Is it 1. A guy who hates women? Or 2. a guy who thinks men are superior. Or 3. A guy who believes in women should follow traditional norms like cooking.
The United States is the only country I know of in which that insult is considered misogynistic, and even then only when it is directed toward a woman. OP is not a woman, and, even though I reside in the United States, I do not consider myself an American. Regarding its offensiveness to women, yes, some women do consider it offensive. If any of them are reading this, then they can go ahead and be offended if they wish, that is their right. But the C-word is not misogynistic. Misogyny involves tools of gendered oppression, the way the N-word has long been a tool of racist oppression. The C-word has never had that sort of power.
If I had called him a “pussy”, then that is something different because it implies cowardice, as though people who own one are cowards. If I had accused him of having a small penis, then that would have been body-shaming and uncool. But cunt? Dick? Asshole? They are just words indicating someone is a jerk, which OP most certainly is.