GJ Group Messages

It sends the message fine now (email or PMS), but it also logs an activity.

Logs an activity entry or logs a notification? It should not be logging either for group messages. What you might be seeing as a PMS notification being logged, which is toggled on/off with CB Activity > Parameters > Notifications > Private Messages parameters. Please check to see if you have those on.

The AutoAction you suggested to me works great for sending emails when a notification is logged on all my sites.
But this weekend, building a new site, it worked for about 2 or three, then stopped sending emails. I can't for the life of me figure out why.
The site is sending mails fine, and CB sends mails fine also (registration), it's just this AutoAction.
Could you take a look? I'll send backend login for you.
Here was your solution (which works great everywhere else):
Repair DB: utf8mb4_unicode_ci
AutoAction:
Global
Triggers: activity_onAfterNotificationEntityStore
Type: Email
User: Code
Access: Everybody

User
if ( ! $variables->getPublished() ) {
    return [];
}

return $variables->getRecipients();

Action
Subject: CB Activity Notification as Email
Body:
This is a test email. Below is the context URL linking to what the notification is about (e.g. activity post, gj group, etc..)
[cb:parse function="getContext" class="var2" /]

Parameters
Format Functions: Yes

activity_onAfterNotificationEntityStore only triggers if a notification was actually stored. If the grouping behavior kicks in that trigger won't fire. My guess is the grouping logic was applied.

Have released a new CB Activity build release adding the trigger to the grouping behavior notification stores. Please update and see if that fixes your issue.

Updated and still the same.
The notification is storing (and appears in the little globe icon & dropdown). 
Also the same AutoAction is in all my other sites and the email is send no problem.
Lastly it was working, for the first 3 or 4 times, then it stopped. 
I must have done something to break it.

I don't know what more to suggest. The only reason that trigger won't fire is if there was nothing to store. In other words it had no recipients. I guess you can just try bypassing CB Activity entirely and handle it with GJs trigger. Below should work.

Triggers: gj_onGroupMessage
User: Code
User:
That triggers variables are as follows.

$row = GJ group
$users = array of users to message
$subject = message subject
$message = message itself

Well in this instance I'm not using GJ, I am using comments on Articles. I may build GJ functionality into this site later.
But maybe there's an article comment trigger? I'll look.