The way web pages work, if your session times out due to inactivity, of course it still says Sign Out in the left hand upper corner.

That is because that was the state of affairs when the page was last retrieved from the server, but then later the server thinks you've been gone for too long and invalidates your last login. Your web page on your computer doesn't change as a result of the server's decision.

Try clicking on the refresh button of your browser (or F5 in IE) to see what happens after you've had a period of inactivity. That should refresh the logged-in indications on your web page. (With an emphasis on should.)