First, the Show Unauthorised Links option is supposed to do the following:
If Show Unauthorized Links set to Yes, every User will see all Article links, whether or not they can access them. In this case, a link to the User Login page will display below the Article Title.
Unfortunately, it does not work that way. If Show Unauthorised Links is set to yes, it shows the whole article. The reason it does this is because unless you have a Read More indicator in the article, Joomla saves ALL article content in the introtext column of the prefix_content table in the database. As a result, intro text is shown when Show Unauthorised Links is set to Yes.
If, however, you have a Read More indicator in an article, then only intro text is stored in the introtext column and all article content below the Read More indicator is stored in the fulltext column. In this situation, only the Intro Text is shown when a public user tries to access an article with an access of Registered. Furthermore, to make it even more confusing, Joomla does not display any message to inform the user that they cannot see the whole article unless they are logged in.
The Show Unauthorised Links option does not work as you would expect. If you have ANY articles with an access restriction, make sure the Show Unauthorised Links option is set to No.