On iOS I need to sign in to a third party account using Sign in With Apple, but do not want to use the Apple ID I use on that device for iCloud. Instead I would like to use the Apple ID I use there for purchases. How do I select the Apple ID to use for signing in Sign in With Apple?
1 Answer
I don't think you can. You have to use your iCloud login
From https://support.apple.com/en-gb/HT210318
You need to be signed in with your Apple ID in System Preferences > Apple ID on your Mac, or Settings > [your name] on your iPhone, iPad or iPod touch. You also need two-factor authentication turned on. Learn more.
and that is your iCloud login (I have the same setup as you a iCloud login and another for buying apps and Music so I can see which one is in Preferences.) The learn More link there goes to your link https://support.apple.com/en-gb/HT211687 which includes
To use Sign in with Apple, you need to use two-factor authentication and be signed in to iCloud with that Apple ID on your Apple device.
and
You need to be signed in with your Apple ID in System Preferences > Apple ID on your Mac or Settings > [your name] on your iPhone, iPad or iPod touch. You also need two-factor authentication turned on.
So this is a service of iCloud and not Apple Id