I can tell you "Jameses coat" is not correct, but I cannot pick between the other two versions. I, for one, have been taught in school that for names ending in "s", the genitival mark is reduced to only the apostrophe, i.e. "James`coat" . However, I`ve frequently found the other version in books, manuals and so on, so I`m guessing they are both accepted forms (for now at least )
People simply do not agree on this and there is no one right answer. I always write "James's."
It seems ridiculous that I would write "Mary's in one part of a sentence and "James'" in another. I would also say it as "james-iz" so the 's make sense to me.
You won't be wrong with either one, but you also will have people disagreeing with you, no matter which one you choose.