How can I check (in the trigger options) that The updater of the ticket IS or IS NOT the owner.
My try was: Owner is not current user.
But this check fails, that means even when I add a note to a ticket I own I will get a mail. (which makes no sense)
Why are you trying to build a functionality that Zammad already provides natively?
By default, a ticket owner will always get updates on his tickets by other users.
Every agent can decide to not receive those mails if unwanted for whatever reason, which they can’t for triggers. You’re potentially “bombing” your agents with emails they don’t want to receive. This may lead to them setting up filter rules to ignore Zammad mails which might be the wrong approach.
Regarding current_user:
This currently is a known bug in Zammad 3.3 ( https://github.com/zammad/zammad/issues/2982 ) which has been fixed by now. It will be part of Zammad 3.4 which is coming soon™