I agree, that would definitely be confusing.
Just to make sure, if you were to enable the JFBCLogin module, and try to login as a new user, does it work properly? I want to narrow down that it's inserting the login button that's causing the problem, and not a configuration issue.
There's no additional code to do what you're looking for as far as registering or logging in. The main setting you'd need to configure to enable the login/register page is setting the New User Creation setting to "Full Joomla User". If you're using the "Facebook User Only", we'd recommend enabling the "Automatically link users by email" setting. This will make it so if a new user clicks the Login With Facebook button, it will create them a new account. If a returning user clicks it, and already has already registered on your site with the same email address as their Facebook account, it will automatically sync them up.
If none of the above help, feel free to post a link to your registration page. A lot of the time, we can look at the HTML output and diagnose the problem.