Here is an example of a previously purchased Kindle book that required Customer Service intervention to receive the latest update:
I purchased the book
Cold Days by Jim Butcher on November 26, 2012. The Manage Your Content and Devices page doesn't show any update available to me for this book.
However, I know that I don't have the latest revision of it. If I use the "Look Inside!" feature from it's web page to view the beginning of the Kindle edition of the book I can see that the bottom of the copyright page has the string "Version_3". I sent the book to my Kindle and when I look at the copyright page there the "Version" string is not present.
Clearly the copy of the book available to me is not the latest. In this case the publisher has put a version number in the text making it obvious that the book was updated. Most books don't have anything like that.
I used the "Live Chat" feature to chat with an Amazon CSR and asked to have the book updated. They did so and now when I send the book to my Kindle it shows "Version_3" on the copyright page, as expected.
ETA: The differences turned out to be a few typos fixed, minor CSS changes, and a teaser for another book added.