The [ link-library ] shortcode is only for displaying links. It's not intended for displaying categories at all.
The [ link-library-cats ] shortcode is only for displaying cats. It's not intended for displaying links.
So if you want links and cats you must put both shortcodes on the page.
I know it confused me the first time I started using link library, because I thought that [ link-library settings=1 ] would simply display everything from library 1. IT DOES NOT! A library is a "merge" of the links/cats/search settings, and the settings=x determines which library is used for that specific shortcode functionality.
[ link-library settings=1 ] means display links and use link settings specified in library 1.
[link-library-cats settings=2 ] means display categories and use the category settings specified in library 2.
There's a whole lot more to it, like using cats overrides and how search results are filtered, but the above should give you basic understanding of how the plugin operates.