Log in and log out using Facebook

Activated "Allow users to register with Facebook" and "Force users to register with Facebook" and tried to log in with my Facebook account (that has the same email as the admin account) this worked fine except for one thing: The login form (username and password) for the standard WP login was still present and I would like to remove that.

Then I tried to logout, my code for the logout link looks like this:

global $user_level;
if ($user_level > 0) {
	echo '<a href="' , wp_logout_url(home_url()) , '">' , __('Log out') , '</a>';
} else {
	echo '<a href="' , wp_login_url() , '">' , __('Log in') , '</a>';

Clicking the link logs me out from Facebook but keeps me logged in on the web site. Subsequent logout attempts just redirects to the home url and does nothing to my logged in status. I am still logged in.