I've seen a co-author on the cover of a book written as "with" instead of "and."

John Jones with Mary Smith

What does with mean? Does it always mean that Mary Smith put it into better English? Or can it mean Mary Smith had some of the ideas?

My friend, Mark, is writing a technical book which will contain adaptations from his first book, which he co-authored. He would like to acknowledge his former co-author, Bob, in this new book, since some of the adaptations are actually the co-author's ideas.

Mark wrote all the words in the first book, but the co-author, Bob, contributed a substantial amount of the ideas, which is why he got a co-author credit on the first book.

Should the second book now say by Mark and Bob?
Or should it say by Mark with Bob?
Or should is just say by Mark (and give Bob an ack inside the book)
Or is there some other way of handling this that you would suggest?

Thanks! Linda