They have to provide it for the Browser to register the cookie if they don't make it available then the Browser will not have it.

The website provides the cookie when I visit it, but I don't want to allow cookies for that entire website everywhere. I want to allow one cookie only on that website, not any website.

You can create a cookie allow exception for each involved domain to keep cookies for specific third-party domains. You need to include the proper protocol (http:// or https://) when you create the exception.

• Options/Preferences -> Privacy & Security
  Cookies and Site Data -> Exceptions

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an "Allow" exception you may want to keep

I can see the option to create an exception, but I only want that exception for one specific domain. I don't see how I can create that. For example, I would want Twitter to be able to use cookies on its site so that I don't have to log in every time, but I don't want Twitter to have cookies anywhere else.

Where is the option to specify the domain that an exception is applicable to?

For what Twitter domain(s) would you require this cookie exception?

If this is for a specific Twitter sub domain then it not very likely that this sub domain is used outside Twitter.

Twitter was probably not the best example I could use. I subscribe to a local news site that uses a third party to manage their log ins and accounts. I only want that third party to be able to use cookies on that local news site, as I'd rather not give blanket access.

You would still have to decide if you want to create an allow exception and have the website remember you.