Depends on D19860. Ref T13222. Ref T10743. See PHI996.
Long ago, there were different types of feed stories. Over time, there was less and less need for this, and nowadays basically everything is a "transaction" feed story. Each story renders differently, but they're fundamentally all about transactions.
The Notification test controller still uses a custom type of feed story to send notifications. Move away from this, and apply a transaction against the user instead. This has the same ultimate effect, but involves less weird custom code from ages long forgotten.
This doesn't fix the actual problem with these things showing up in feed. Currently, stories always use the same rendering for feed and notifications, and there need to be some additional changes to fix this. So no behavioral change yet, just slightly more reasonable code.