I was finally got it all to work.
Inspired by the link @MrGeneration gave about configuring a sendmail local MTA, I came up with this to instead add an Email Account without incoming, but with outgoing configured to use our SMTP server:
rails r "Channel.create(area: 'Email::Account', options: { inbound: { adapter: 'null', options: {} }, outbound: { adapter: 'smtp', options: { host: '<HOST>', port: 25, user: '<USER>', password: '<PASSWORD>' } } }, active: true, preferences: { editable: false }, updated_by_id: 1, created_by_id: 1)"
This gave me
I added the Email Address to it manually using the web UI.
I then configured this Email Account as the account for all our ticket groups.
Then I could configure the following trigger:
And fill in the following template for the mail:
Hello #{ticket.customer.firstname},
Your ticket at <OurCompany> Support has been updated by #{ticket.updated_by.firstname} #{ticket.updated_by.lastname}.
Please visit #{config.http_type}://#{config.fqdn}/#ticket/zoom/#{ticket.id} to see your ticket.
Customers are now notified the way we want them to be when there are updates to their tickets
Many thanks for your suggestions @MrGeneration and @rsysadmin, they both helped out in the end.
The only minor annoyance I have is that on the ticket, Zammad prints out a text
"Your ticket has been updated" -> "e-mail@of-the-customer.com"
each time the trigger runs, and even customers get to see this text. We would like to turn off this printout completely.