the place where his use of these terms differs from everyone else is that he thinks he has to add all sorts of abstractions about judgements, when in fact how everyone else uses the word "evidence", the judgement phase is already included.
I think what he's been implying but not saying directly is that when he "judges something as evidence", it might not really be evidence. Like, to him, it's only REALLY evidence if the thing it's evidence for is also true. It can be judged as evidence, but only mistakenly, if it's not true.
But to everyone else, there's no difference between judging something as evidence, and evidence. Those mean the same thing.
Like, if you say "I believe X" and I say "why? What's your evidence?" you might say "my evidence is this this and this, but I'm still not certain" while his vocabulary would force him to say "I have judged this this and this to be evidence, but I'm still not certain". Nobody else needs to add the word "judged" in there, it's already implicit.
Perhaps another way to phrase it is, for him, "evidence" can only be objective (evidence can only be objectively true signs of objectively true facts), so if he wants to talk about subjective reasons for believing something, he has to add the word "judged" in to subjectivify it. But to everyone else, "evidence" already has that subjective nature implied.
And of course it already has subjectivity baked in. Objective truths don't care about objective evidence, evidence is how imperfect human beings share ideas about their uncertain beliefs. The only reason "evidence" is a word at all is because it's useful in cases where humans are sharing their judgements about their uncertainties, and to convince other people of their conclusions. We don't need to add words to subjectivify "evidence", it's baked in.
Courts say 'present your evidence', not 'present what you have judged to be evidence'. To native English speakers, "what you have judged to be" is redundant.