I think the problem is that private browsing won't save text box inputs. Deleting my saved Pprune login details, opening a Pprune link in a private window & manually logging in gets me in but the username & password are not saved in the Firefox password manager. You can manually add a website & login details in the password manager & that will be saved, even in a private window.
Your alternative of saving logins in a normal window will also work, private windows have access to all saved passwords. You won't be auto logged in though as that requires a cookie & private windows won't save them after the browser is closed. Click on the username field & Firefox will find the relevant details.