Detail like websites remembering you and log you in automatically is stored in cookies, so you need to clear these session ID cookies to make the website forget about you.

• using "Delete cookies and site data when Firefox is closed" to clear cookies keeps cookies with an allow exception
  in 102+ version toggling this setting makes changes to the "Clear history when Firefox closes" settings and those settings prevail
• using "Clear history when Firefox closes" in Firefox 102+ honors exceptions and keeps cookies with an allow exception, previous versions removed all cookies

Make sure to keep the "Site settings".

• clearing "Site settings" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"
• Settings -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings

TY. This is a bit advanced for me, but I still don't think it resolves my issue. If I tell Firefox not to save passwords, then why would cookies be allowed to do so? I assume you have cookie rules for Websites. Isn't there a way to not allow cookies for passwords to be saved? I do not want to make exceptions, I just don't want passwords saving.

Reputable sites like Google do not save your password in a cookie. They may save your user name as a convenience to you, and they save your current session token so that you can stay signed into your session until you sign out on the site.

Firefox lets you control cookies on a site-by-site basis (enabled/disabled) but it doesn't filter among different kinds of cookies from the same server (user name, session token, preferences, etc.).

If you like, you can set Firefox to clear your Google cookies when it closes. You can do this either by having Firefox clear ALL cookies when it closes, or by expressly clearing Google cookies using an Exception. Here's how:

Open the Settings page, click over to the Privacy & Security panel, and scroll down a bit to the Cookies and Site Data section:

(1) Clearing All Cookies at Shutdown

There is a checkbox here labeled "Delete cookies and site data when Firefox is closed".

If there are any sites where this is inconvenient, you can use the "Manage Exceptions" button to create an "Allow" permission so Firefox preserves cookies for that site.

(2) Force Cookies to "Session Only"

If you prefer to keep most cookies but want Firefox to clear Google cookies, click the "Manage Exceptions" button and set the permission for https://google.com to "Allow for Session." This applies to future cookies; any cookies already set can be cleared using the "Manage Data" button.

Note: On some sites, "Allow for Session" may create issues saving local data, so if, with this in place, the site tells you that you need to allow cookies, then unfortunately this method won't work for that site.