If after this you keep being logged in again without going through the login page... more than likely your browser is Internet Explorer.
There is a known issue concerning the logout with this browser.
To be able to end your session properly:
- First, use the logout link on Statigram (click on your profile picture on the top right and select Logout from the drop-down menu)
- Then, to completely close the session, click on this link: http://instagram.com/accounts/logout/
Alternatively, if you can, change browser. Use Firefox, Chrome, Safari, etc.