It should appear a second after the page seems to have finished loading, so that it hijacks the input if the user is navigating with the keyboard or typing something into a form.
LinkedIn today is so serious about its cookies it can't do proper logout. As long as there's a cookie, I don't know if it's random or what's going on, but if you log out... there's a > 20% chance if you leave for a while and then go back to LI... you're still logged in!
If you log out, and then (in Firefox) close the window... oh wait, you're not really logged out. Try again.
If you log out in Firefox, wait until the window is quiet, then delete the cookie... LinkedIn creates a new cookie... which still sometimes logs you back in but not as often.
Opening a new window, deleting the LinkedIn window (fuck watever it's doing), and THEN delete the cookie in the NEW window... then it seems to stay gone at least as of today.