After writing that, I decided to check a dictionary. This is how a web page from Collins Dictionary puts it. "If someone who is involved in an argument or dispute has behaved in a way which is morally or legally wrong, you can say that they are in the wrong. 'He didn't press charges because he was in the wrong.' Synonyms: guilty, mistaken, at fault, off course"

"Who's wrong?" is more commonly used in spoken English than "Who's in the wrong?" "Who's in the wrong" would be more commonly used for formal writing, perhaps reporting in the news or in educational material. I believe it would always be better to default to "who's wrong".

I agree with Dan. I just looked it up. "Whose in the wrong" is used to describe who is morally or legally incorrect, when someone is mistaken, or is to blame. Such as "She is clearly in the wrong. She had no right to take that book." I hope that helps.

They are not quite the same. "Who's wrong?" is common. It often refers to a question of factual accuracy. "Wrong" means "incorrect" or "mistaken" or "inaccurate." "He says there are nine planets, she says there are only eight. Who's wrong?" Of course, the question can be something deep or religious or philosophical. "Who's in the wrong?" is rare. It often means "who is at fault? who did the wrong thing?" It involves a judgement of bad or good behavior. For example, if two children are fighting, you might ask "who's in the wrong?" Which one started the fight? Were they justified?