Good morning Pverburg!
I see that you're having some trouble logging in to iTunes on your Mac after logging out.
First, make sure that the issue isn't with the Apple ID and that you can sign in successfully with both the Apple ID and password here: Manage your Apple ID - Apple. This will rule out any issues being with the account. If you're not able to log in, follow the prompts to reset your password then try logging in again on the Mac.
If you're able to log in successfully using the web browser but still not able to in iTunes, let us know the error message that you receive when you attempt to log in.
Take care.