Apropos of nothing - a random HP thought.

Where does half the fandom get the impression that Harry is cheating by using a book with notes in it?

They are allowed to use their book while brewing and are supposed to have read up on the potion they are preparing in class.
And in written tests, I doubt they are allowed to use their books anyway, so the notes won't be any help to him there.
The only difference I see between Harry and a Ravenclaw (in this instance) is that the Ravenclaw would have made the notes himself - and probably not in the book, but on a spare bit of paper.

Having better reference material is rarely considered cheating. No matter how much of an advantage it gives you.