Even if private browsing is deselected, 2 options remain operational beneath it for setting cookie choices: whether to accept third-party cookies, and how long to keep cookies.
Under "Firefox will:" there are three choices:
this "Remember history" I already mentioned,
"Neva rememba history"
and "Use custom settings for history".
Was it merry?
Did you marry Mary?
OK, in my FF setup, I've selected "Use custom settings for history". I'm not sure what the 'default' setting would be, since on my setup, none of the choices carry that particular label. The other choices for that selector are Never Remember and Always Remember History.
When using that custom settings option, the following checkbox options appear beneath:
Always use private browsing mode
Remember my browsing and download history
Remember search and form history
Accept cookies from sites (followed by a choice box for Exceptions)
Accept third-party cookies: (followed by a choice box for Always, From Visited, Never)
Keep until: (followed by a choice box for They expire, I close Firefox; and another choice box for Show Cookies)
Clear history when Firefox closes (followed by a choice box for Settings)
It may be that the custom settings option reveals the cookie options beneath. It's been a long time since I set this up, and Firefox just keeps inheriting the settings as versions have been updated. I don't want to take the chance of changing the custom settings option in order to find out, just in case that might dump the subsequent cookie choices I've made (especially my lengthy exceptions list).
PS: Mary was already married to a guy named Joseph.