Try if this works now. Some constants of ZOs for the group texts have changed:
GroupNotifications_Baertram_Dragonhold

I've also changed the texts to always show charactername link (@accountName) -> Except for group leader change as there is no easy parameter for this.

Edit:
Found out what is the problem with group join: ESO...

First event group member joined wil lfire with YOUIR OWN charactername only.
And all others with the other new members added.
So if you invite to a grup the first toon you won't see any message or your own character name only.
Cannot change this, this is ZOs made "logic"...

One would need to check if the group size is 2 and then implement another logic like "loop over all group members, strip the one which is the player and show the other's name then to chat" to fix this

Not working. Now I'm back to not receiving anything in chat about players joining or leaving group.

I've tested this with minimal addons enabled to check for conflicts. Conflicts isn't the issue.
(Enabled addons & libraries: Addon Selector, Group Notifications, Votan's Settings Menu, LibAddonMenu-2.0, LibDialog, LibMainMenu-2.0)

Edit/Update: Ahh, just saw your updated comment. So it's only going to work properly after the 2nd member joins?

Yep, trying a fix like described but cannot get the displayname for that char then I guess.

I can be happy with it working properly after the 2nd person joins. Because my main concern is with big groups. Like 12-24 members. So if it does in fact work correctly (joining and leaving messages) after a 2nd person joins, then that's not a huge problem in my opinion. I appreciate you looking further into it, though.

Okay, last try. Should fix message if first group member is invited (only possible without @displayName though) and kick messages as well:
GroupNotifications_Baertram_Dragonhold

IT'S WORKING!!!!!!!!
Thank you, Baertram.
@CaptainBlagbird - please consider uploading Baertram's version as an update.

@CaptainBlagbird - I've been using the version Baertram worked on for 6 days now, and it works wonderfully. Please post this version as an update.

Thanks CaptainBlagbird, will do so

I made a forum addon request on ESOUI, but I'm sharing it with authors of group addons...

So... I lead a lot of farming groups. Especially dolmen farming groups in Alik'r.
Apparently, a new trolling trend is to "ready check" randomly, and sometimes as much as possible.
It's maddening.

Since ZOS didn't give us the option of only allowing group leaders the privilege of doing ready checks, we could really use an addon that notified us in chat who started a "ready check", so we could then weed out the trolls and kick them.

Check the answer in your forum post please.
I've added link to the FCOChangeStuff beta providing a setting + keybind there.
I'm not going to add this special settings to this group addon here as the only idea of this addon is notifying you.