How to switch users in Twitter after logging out?

I am currently working on my Voting app, my login link currently points to http://127.0.0.1:8080/auth/twitter.
I can logout, but whenever I log in I always get logged in automatically to my previous twitter username (presumably due to saved twitter cookies?), what if I want to log in as a different user? is there a different link where it’s possible to log in as different user after being logged out or should I clear the twitter session cookies?

It’s supposed to work like that. If you want to login as a different user you need to make a logout on twitter website or clear cookies.