Welcome .
The scenarios when changed -
To change to a different iCloud email address, you can't add an email address or change an email address. You need to create a NEW iCloud account.
Not necessary you can add an email address in Mac system preferences > internet accounts , you don't need to create new iCloud account , these mail addresses were already created from other device iPad / iPhone or other Mac , but they can be utilised in your primary Mac , the primary Apple ID of your " Mac A " will be still signed and working .
Use your Internet accounts on Mac - Apple SupportUse your Internet accounts on Mac – Apple Support (AU)
Change Internet Accounts preferences on Mac - Apple Supportsupport.apple.com › en-in › guide › mac-help › mac
Add an internet account - Apple Supportsupport.apple.com › guide › mac-help › mac
You keep your primary one, and then just use the email address associated with your new account.
Yes the primary Apple ID is still signed in , you can use other mail address either fromweb account ( like signing in from gmail server if apple id is created as example2@gmail.com or what ever service you had created .