which version are you using of the plugin ? see it in the admin/plugin/"open login" panel.
I have the 2.0.3 installed, a dev version, I think you have the 2.0.2.
But I do not think this is the problem.
I do not either see the code of the button appear in your web page....
Are you sure you have checked the "Enable LinkedIn" option ?
If you exec this query on your database :
SELECT * FROM `qa_options` where title like '%linkedin_app%' or title like '%open_login%'
what's the result ?