I distrust Shakespeare's version of the events -- it's good literature but bad history. He wrote at a time when the concept of a nation was full-blown while during the Plantagenet wars the concept of the nation was not so distinct.
Writers take famous names and elements of stories, throw in a dash of popular sentiment and their own prejudice, and voila! Historical drama. Shakespeare knew his audience (including a certain Tudor queen) and just did his best to please them. And his best was very good (drama).