"The cookie will be saved and returned to your script until this expiration date is reached ,or if the user exits the browser and restarts it."

That is so untrue. Under normal circumstances, the browser saves the cookie to disk, and it will be used until the expiration date has passed — unless the user deletes it, of course. Restarting the browser, or the computer, won't make a difference.

If you want a cookie to last only until the browser is closed, don't provide an expiration date.

p.s. An expiration date in the past will delete the cookie.