Distributed notification

A macOS distributed notification is a named message sent between processes. An application can broadcast a message, optionally with an object and additional data, for other registered processes to receive. It is not a banner in Notification Center. Apple's DistributedNotificationCenter documentation

BetterDisplay can use these messages in two directions. Under Application Integration, it can listen for incoming integration requests. As a Control Integration action, it can send a notification to other software.

Configure the direction you need. Enabling BetterDisplay's listener does not create a receiver in another application, and sending a message does not prove that another process received or handled it.