Mmm...it's tricky.
Generally speaking, I'm sure most people would be happy to click a title of a book they have read, without having to worry about the edition. Readers may recall books from years ago, but not the edition or publisher. And there will be those who read a book and then pass it on, or view digital editions.
How an 'I've read' button could be put into practice, I've no idea - although there seem to be some very capable technical wizards on this site!