I don't know what the ethics rules say for refs, but in the legal system bias and apparent bias are almost always lumped together as effectively equivalent, for the exact reason you can see with fans now scouring over Coote's entire record of decisions.
It's impossible to read someone's mind and tell whether a decision was actually because of bias or not. So, when a clear appearance of bias exists, it is all basically the same from an outsider's POV. Every single decision is just as much a candidate as the other to be tainted by bias.
And that's not an indictment on the fans for being paranoid, because (again) they have no way of reading Coote's mind and knowing which decisions he made because he doesn't like Liverpool. All they can ever know is that potentially any of them could have been. And that's why professional organisations, which need their decisions to be taken seriously, take the appearance of bias just as seriously as bias itself.