A proof is a page that is almost finished. The proofreader’s job is to make it finished without anyone noticing that it wasn’t.
A small vocabulary of marks
There are marks for a missing space, for a letter turned upside down, for a word that should move one line up. Most proofreaders use a dozen of them and invent two or three of their own.
The red ink is not for drama. It is simply the colour least likely to be mistaken for print.