The first link is a "single post" view of a post that is part of a series and thus showing the box for linking to other posts in the series (and the navigation links for going to the next or previous post in the series)
The second and third links are to "archive" pages which typically are used for showing multiple posts in a category (or in this case a series). By default Organize Series doesn't automatically insert the series box on archive pages because typically people use archive pages to list the posts in a category, etc. With your current theme, it looks like you've set your archive pages to only return one post per archive page (and the full post at that).
You can still show the series box and navigation links for the post on archive pages but not via the automatic insertion provided out of the box with Organize Series. Instead, you'd have to modify your themes files and add the organize series functions for displaying series table of contents and navigation manually.