To clear cookies only from this website follow this suggestions:
(please note, that I am using Chrome browser, similar steps work with Firefox browser)
This topic it’s about teaching you how to clean the cookies for CGC only
Not necessarily clean all the cache and history
Step 1: click on the lock visible on address bar (before https)
Step 2: click cookies
Step 3: select the site url
Step 4: click delete/remove all
Deleting cookies from CGC might be a solution for enabling your login with Twitter account