The setup I use is not sophisticated by HN standards, but it does not in any way rely on the application, e.g., the web browser. I do not use the browser itself to try to control the browser's activity. I do not use browser settings, dev tools, extensions, etc., which seem to be the more popular choice.

As I mentioned in the original comment, some websites, like banks for example, will log the user out after a period of insactivity. Whereas tech companies, always trying to collect more data, may encourage users to "stay logged in", ideally forever.

Having session cookies expire could be an option that users could control. The user could choose to have the site log her out after x minutes of inactivity, or some such. The site could agree to "expire" the session cookie on the server side such that it could not be used again. How would the user signal her chosen option to the site. I leave that to the reader. Perhaps an HTTP header sent with the initial log in request: Session-Cookie-Max-Age: [seconds]