How can I completely de-authorise my account from iTunes (mac)? I'm using a different apple ID now.  So I switched accounts like this:
- I signed out of the old apple ID in System preferences > iCloud.  Then signed in with the new one.
- I signed out of the old apple ID in App Store > Store.  Then signed in with the new one.
- I de-authorised the old apple ID in iTunes > Accounts menu > Authorisations. Then authorised the new apple ID.
I restarted the computer.

However, even after authorised the new Apple ID, it's the old one that still shows up in iTunes > Account, on top, even though it has already been de-authorised:

Screen Shot 2017-11-05 at 16.56.55.png
not only that, now, when I go to Account > View my account, I'm prompted for the old apple id and password, and if I use instead the new one, it won't go through, I'm prompted again for the new apple id and password.

Weird!.
Signing out of your account and deauthorizing a device are two different things.

Within iTunes you can actually sign in and out of multiple accounts. I don't mean simultaneously, just that you can sign into one and, if needed, sign out of that account and then sign in using another account.

Authorising a device basically tells the iTunes ecosystem that it's allowed to play your purchased music, videos etc and that's why there's a limit to how many devices can be authorised for any particular account. So, deauthorizing a computer/device is typically something you would do when you're going to sell it etc.

If you want to sign out of your old account in iTunes you should actually do that within iTunes. You may need to sign in again first and then sign out before signing in with your new account.