A failed login attempt on the upgrade page exposes the POST action (admin.php?login/login) and takes you to the Admin CP login form. At this point you have left the upgrade page completely. A successful second login attempt redirects you to admin.php?login/login via GET which further confuses the problem with a 'This action is available via POST only' error.
The upgrade login form shouldn't take you to the Admin CP after a failed login attempt. It should keep you on the upgrade page for successive login attempts.