I FOUND IT.
This is actually not in the documentation for individual Custom Anim Notify. The section for Notify State helped me figure this one out.
If you open your custom AnimNotify BP, you can override the function ‘Received Notify’ and enter your logic in there.
This is why it does not show in your Animation BP’s EventGraph, because you have to enter your logic in your Custom AnimNotify BP itself by overriding this function. NOT in the Anim BP EventGraph where you receive events for your ‘New Notify’-created ones. Refer to main post to see that. Big difference between the 2.
As for the triggers triggering twice, this was entirely a networking issue on my part. My networking logic was wrong and it was resulting in 2 AnimNotify calls being called over the network from the client and from the server, so 2 at once.