share-this is for any of the sharing options available on the site. That doesn't just include the permalink with your referrer ID on it, but also the Facebook, Google, and Twitter buttons that appear in that box, at the bottom of a question asking if you know someone that can answer, and possibly other places that I'm not familiar with.
permalink is specific to only the permanent link that is accessible to every single post on the site. It uses the post's ID and redirects the view, and is meant to be a URL that will always work and never need to be changed no matter how the question gets edited to change the title.
short-links is for the http://s.tk short links that provide shortcuts to various key pages of the network. It is not related to the sharing or permalink features in any way. See the FAQ: What shortened URLs are available through s.tk?
I've updated all of their tag wikis with relevant information for each.