Following query is the proper FQL Query for retrieving like or comments count for a link
select url, like_count from link_stat where url="https://ballentines.herokuapp.com/?name=charlatan"
I have debugged the above Query with Graph Explorer
Reference : https://developers.facebook.com/tools/explorer/145634995501895/?fql=SELECT%20like_count%20FROM%20link_stat%20WHERE%20url%20%3D%20%27https%3A%2F%2Fballentines.herokuapp.com%2F%3Fname%3Dcharlatan%27
If you check on above link, you would get proper like_count (now 3). Screenshot below
If you check above Screenshot the application selected is "Graph API Explorer" which is the default App while debugging with Graph API explorer.
If you are logged in with your proper Developer Account. the application Dropdown would also have your Application.
So after switching to your Application, click "Get App Token" and then submit the Query.
Now the like_count would come 0 for the same query. See Below Screenshot.
Now click on "Get Access Token", this would ask you to Allow the App if you not already done so. After "Allow App" proper User Access Token would be loaded in "Access token" box and now again Submit your Query. Now the "like_count" would be 3 as expected.
So you would need proper User Access Token to retrieve proper information from "link_stat" FQL table and App access token wont work for it
Hope this helps.