Let New Users Connect with Twitter OAuth only (not Facebook)
You can look into HybridAuth Social Login


  This module integrates HybridAuth library into Drupal and allows your
  users to login and register using:
  
  Facebook Twitter Google LinkedIn Foursquare Yahoo Windows Live MySpace
  AOL OpenID Github, LastFM, Vimeo, Disqus, Instagram, Viadeo, Identica,
  Tumblr, Goodreads, QQ, Sina, Murmur, Pixnet, Plurk, Skyrock, 500px,
  Vkontakte, Mail.ru, Yandex, Odnoklassniki, Twitch.tv, Steam and more


You can select which all social logins you want to enable. So if you are just looking for twitter you can enable that alone.