No "ticket updated" email notifications received by client?

No "ticket updated" email notifications received by client?

With an existing ticket, when an agent updated the ticket, I'd like the system to send an email notification to the client and let him know that the ticket has been updated. That's all. Sounds simple but I have not been able to figure out how to do it. In Setup-->Notify, there are only notification options for "public comment". My issue is that when an agent responds to an ticket via email, it is treated as "reply", not a "comment". Hence no notification is generated. My ask is simple - How can I have the system send email notifications to the client and the agent when there is any update to the ticket?

A user already receives ticket opening and closure emails, why he can't receive ticket status update emails? I must have missed something simple...