Thanks for the response. I did not change any passwords. I changed the security setting as Yahoo advised
What security settings did you change? Did you turn on two-step verification or Account key? If so, then you need to will need to create an "app password" (you basically have to create separate passwords for specific apps). You can find instructions here:https://help.yahoo.com/kb/mail-for-desktop/create-password-sln15241.html
My Apple app still tries to sign in. This is the message in the response to the attempt:
Your account is currently not enabled to sign in from apps, that do not meet modern security standards (ex. Older versions of mail and calendar apps such as Outlook). As a result, we prevented a sign in to your Yahoo account (xxxxxxxx).
Date and Time: Mon, Jan 2, 2017 7:50 AM EST
Location: United States (IP Address: XXXXXXXXXXX)
We strongly recommend that you switch to Yahoo's apps such as Yahoo Mail on desktop and mobile and remove your account from all other less secure apps.
If you still want to use an app that uses less secure sign in, go to https://login.yahoo.com/account/security#other-apps and turn on "Allow apps that use less secure sign in". This is not recommended and may leave your account more vulnerable to compromise. To learn more, please visit our help page: https://help.yahoo.com/kb/account/SLN27791.html
This second part seems to imply that is just Yahoo blocking access for older version of Apple Mail. For more information, read the help page in the second link in your post. From there, Yahoo considers Apple Mail on Mac running macOS 10.9 or older to be using "older security protocols" and blocking access. You said you are running macOS 10.8.x, which would fit that "older security protocols" cutoff.
If it is this second reason, then from that help page that the second link in your post goes to, your options are to use the webpage mail access, upgrade macOS to at least 10.10.x (if you can), or switch to a third party email client that supports "modern sign-in security" (the Yahoo help page suggest newest Gmail or Outlook mobile apps, which does not help...but the latest version of Microsoft's Outlook for Mac might work or maybe Thunderbird). That help page also does offer a way to allow mail clients with older security protocols to sign in "temporarily" as well, but Yahoo specifically states that they don't recommend that...and they don't really state what "temporarily" means.