Sharepoint 2013 and browser logout issue

I'm not very familiar with this but I'm not sure I understand what you are saying - the options for cookie persistence appear to me to be creating affinity with the backend servers rather than terminating the client sessions. What am I missing?

There is a setting within Chrome browser (default) (and Opera) that sets Cookies option to "Allow local data to be set - recommended". If I turn this off, the browser behaves as expected and terminates the session. As we do not have control over many of the systems that our students use to access our Sharepoint installation, I cannot change this default behavior, so I was looking for a way to do it through F5 as it is done through TMG where you can set the option for using persistent cookies to "Only on private computers"