For your staging site, I was able to get a user created with the automatic registration properly after your new change. When I clicked Facebook login button, the popup to approve your app appeared. Once I selected confirm, the user was created, the "Facebook profile successfully connected." message was displayed and I was logged in properly. You can see the user with my "Melissa Lynne SourceCoast" test account. I was also able to log out and log back in again just fine.
* Make sure that the FB user that you're attempting to log in with does not have an entry already in the JFBConnect > UserMap table.
* Check whether the email address on the FB account matches an existing Joomla account. If it matches and you don't have the 'Automatically Link Users by Email' option set to Yes, then you'll get the login / register view that you're seeing to enter the new user details.
* Can you try with a different Facebook account?
For Google, there is an error in how your application is set up since I am getting an error about the redirect_uri_mismatch. However, as you noted this is probably since you are using the staging site. If you'd like to test the Google authentication on your staging site, please make sure to add its appropriate redirect URI values as well. You can refer to our
Google Application Setup Guide
to verify you've set this option appropriately.
-Melissa