I have an art book under my hands, but am, because of my curiosity, struggling to understand the difference between:
I thought the first two and last two were the same thing. Are they not?
Disclaimer: I am not involved in book publishing and therefore these definitions may not be technically correct. They are based on my lay-person's understanding.
Designed: Who prepared the visual aspects of the book, the illustrations and general design.
Produced: Who oversaw the rest of the production of the book, typesetting, arranging pages and producing the file for printing. This is a rather unusual phrase if you ask me (British English speaker) and I haven't really encountered it applied to books before.
Published: Who has the rights to the edition (not the content, as that has a separate copyright), who hired the designer, producer and printer, who oversaw the whole process of getting the book from someone's head onto the paper that you are holding.
Printed: Where the book was literally put on paper, where the pages were printed.