Persistent Internet Account Entry in Keychain

In your case if second Apple ID would be there say example2@icloud.com it would be shown here in internet account ( in case if its not deleted by clicking on subtract sign ) , but when new Mac is set up definitely system preferences , Internet accounts will appear essential an Apple ID is signed in .

You deleted an old Apple ID say it was example1@gmail.com from the server , to set up gmail account a user name its password , its secondary mail address , date of birth .....in gmail account is required , see the link https://support.google.com/mail/answer/56256?hl=en

In it is mentioned the same user name that was previously used can't be repeated ( google doesn't know you are the same person that is creating the same account ) .

It depends what credentials of the old account you remember .

In internet accounts , click on inactive account , check mark the mail box .

A pop up appears click on open safari .

The account is permanently erased and can't be recovered .

But , in your case I can't see any Apple ID like example2@icloud.com or what services you use , in Mac system preferences > Apple ID is there any Apple ID signed in , or you are in process of setting up new Mac as per apple article , do other iPhone , iPad / other Mac still synced with the account ?

You have erased the google account say example1@gmail.com from their server , that's the reason its showing inactive in internet accounts , however it was an apple id that was signed from Mac system preferences , but the Apple ID is still active on iCloud servers , when new Mac is purchased , you used the same Apple ID and password to sign in the iCloud Keychain won't be set up , as primary mail account is erased from the server , that has to be first recovered .

See some articles Set up your MacBook Pro - Apple Support

Set up iCloud Keychain and iCloud Passwords on all your devices - Apple Support

And a thread https://discussions.apple.com/thread/252994290

Note : if you contact apple support , don't delete the account from the iCloud servers , as if it's being erased the account can't never be recovered ( after 30 days account is permanently erased ) How to delete your Apple ID account - Apple Support

Let that item remain in the keychain access , even its inactive right click on item click on delete ( its just for information ) , keychain access is very sensitive , check mark show passwords it will prompt to enter admin password View the information stored in a keychain on Mac - Apple Support

Read What is Keychain Access on Mac? - Apple Support

When Mac OS is installed in between process it asks for your old Apple ID address say example1@gmail.com once the gmail account is erased permanently from the server the same Apple ID can never be used to sign in activity .

Sign out / sign in has nothing to do with it , it will not help , even you set new admin account .

Initially what I thought the account can be recovered , I had deleted an old account that what is in screen shot , there is time limit of 90 days to recover the account , after the time limit it can never be recovered , in the same way it applies to your Mac .

You are using an Apple ID that is signed in Mac system preferences , so just use it but forgot about erased account from the server .