FB Like after logging out

With that setting you mention, you should be logged out of Facebook as well as your Joomla site when you log out. In that case, the Like button should still 'work', but it will likely ask you to login to a Facebook account before actually 'Liking' the page.

Can you go to Facebook.com after you logout of your site and check if you are really logged out on Facebook.com too? That will help narrow things down.

Thanks,
Alex