"Amendment" or "Addendum" for book correction

Solution 1:

Errata would be the term for corrections of errors. I think that your suggestion of amendment would be a better choice for a clarifying edit of the original document, assuming that it did not contain outright errors.

Solution 2:

Sounds like you're looking for corrigendum (pluralized as corrigenda). Also,wiki lists errata as its synonym.Both refer to corrections made after a book has been published.

Solution 3:

You indicate "a separate correction for the book is released". A separately published section is generally a supplement, "a publication that has a role secondary to that of another preceding or concurrent publication ... either by bringing it up-to-date ... or by otherwise enhancing the predecessor's coverage of a particular topic or subject matter...".

Previously-suggested errata or corrigenda ("lists of errors and their corrections [in] form of loose, inserted sheets or bound-in pages") and addenda ("addition required to be made to a document by its reader subsequent to its printing or publication") are specific forms of supplements, a term which is slightly more general.

Appendix typically denotes "a text added to the end of a book or an article, containing information that is important to but is not the main idea of the main text" and does not specifically connote separate publication.