Right. I haven't really explored any of the "private browsing" settings in any of my browsers. What I did find was I could set Chromium to accept local cookies, log in to LastPass, then log in to sites I visit the most. Close Chromium to save the cookies. Then change the setting to Accept Local Cookies - Session Only. That saves me from manually logging in to LastPass and the sites in my Speed Dial. Now and then I've used ChromiumCookiesView
to check if any tracking cookies have gotten in. Seems to work well.
For Firefox I can set LastPass to log in on start without a cookie. I just set it to delete all cookies on close. With Opera, delete new cookies on close. Seems to work fairly well. Trouble with filters is they must have a way to read the filter and change the cookie name to get around it.