Thanks for the reply...
Yesterday DayZ Server was not showing up under Tools at all.
Now it is, and SteamCMD works as well. I guess there was just a big delay until DayZ Server got linked to my Account for some reason.
Problem "solved".
When trying to download the server files using SteamCMD, it returns:
And yes, I am logged in using my account which owns DayZ.
Okay, this is strange, when I look at the DayZ Server in the Steam Store I'm getting this: