Have you found this session stored (I'm presuming you mean some sort of session ID as the session looks to be every event you do during your time on a page - before something loads but I might just be confusing my javascript knowledge with this). I spent a couple of minutes fiddling with the console trying to find a session Id and have failed so far - the cookie for login will be there though.
I always thought the mechanics were that the cookie stored a number on here and the server stores a cooke number that grants access to the user account when you send the number at the start of a session, I thought this number was used as the session ID (not sure on this though) normally when developing sites - on assigments - you just keep a session ID of the user (which could be what is stored by the server) and this ID doesn't go when you refresh the site and only goes once you've shut down the browser (but this isn't using javascript, it's normally using backend server programming).
The 60 minutes thing is definitely right and I skipped past that thing as I read the thread going down, if you don't tick the box and you get logged out on your new session - all of your sessions will get logged out as there is only one cookie stored [a session in this case is the cookie ID which just gets rejected by the server and has to be overwritten when you next log in).