Even in Private Window mode, after having logged out from an account and then having refreshed the web page (Issue currently occurred with the Google GMAIL service), had no effect on the login display as the page kept displaying the previous login name. Issue could only be avoided by shutting down the current tab and reopening a new one with the same login page.
Private window means that when you close that window, everything from that window is forgotten. If you didn't close the window, then it isn't forgotten yet. Hence, I don't see any issue.