iCloud sign in issues on Mac

Your iPhone and Mac are connected with same Apple ID and password , in your iPhone use the original primary Apple ID to again sign in https://support.apple.com/en-us/HT204053

In Mac , in your screen shot in system preferences > you can click on cancel , now take cursor in the dead end corner and delete the Apple ID , there must not be a single character in the field , use delete button of your keyboard .

Now you have to sign the exact Apple ID and password of your iPhone ( it depends what account was created to sign in , the secondary authentication account or 2FA ) , the steps will vary for both accounts .

You don't need to go to system preferences > internet accounts and delete them by clicking on subtract sign .

I'm not sure how you swapped the primary address for the other associated address, I didn't think you could do that, if indeed that's what you done. Nonetheless, here is a guide for changing the email address for an Apple ID.

https://support.apple.com/en-us/HT202667

You didn't touched the settings in iPhone ( the Apple ID is signed in iPhone ) , what you did gone to Mac opened system preferences > iCloud and signed out and then tried to sign in with new mail address from system preferences .

As you said - Can now no longer sign into iCloud on my Mac, as Mac says I’m signed in already. ---it is correct the iPhone and Mac are still connected and the devices are not separated from Apple ID account .

You mean before I changed the email address on my primary Apple ID i should have signed out of iCloud on everything?====yes see the article https://support.apple.com/en-in/HT208242

Can not sign out of iCloud on phone, username / password no longer recognized. ----the iPhone is still connected to the Apple ID account .

Seriously I never tried this way of signing out , the devices .

Right now I would suggest contact senior apple care advisors in apple support https://support.apple.com/en-us/HT201232

That is sure a way can be found out with out resetting iPhone and reinstalling Mac .

Is iPhone and Mac still signed in with same Apple ID and password say example1@icloud.com on the same network .

You only sign out iCloud from Mac , and then deleted iCloud account or gmail account or both of them from Mac system preferences > internet accounts ?

The iPhone is still signed with Apple ID and password ( it could be using some mail services ) , in Mac click on cancel button ( in your screen shot of Mac ) what exactly do you see is Apple ID showing with password .

What is the mail address signed in iPhone ---- which mail services is used , post back like say example1@gmail.com

In Mac , when you click on cancel ---which mail services is used ( say example2@icloud.com ) don't post your real mail address .

I am not sure if terminal commands are permissible , the method what I suggested was only manual I applied in a phone and Mac .

The Apple ID was signed in iPhone , in Mac you signed out , but it was not allowing to sign out as it was connected with iPhone ( synced with Apple ID ) .

Please post that terminal command here what exact steps were followed .

I have same issue here my AppleID was Hotmail email and I changed it to iCloud email and it behave same as "port_mobile

" mentioning above..

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and no ohmto@icloud.com in the internet accounts

If the gmail address and the Apple address are both associated with the same Apple ID, then you should be able to use either to log in.

And have you tried removing it. If it's the same Apple ID, it won't want to be added twice.