Hey neighbors! Thank you for sharing your experience with the Community. After looking over these scenarios, and testing them internally with our teams, the intended functionality is as follows.
For powered Doorbells (Doorbell Pro/ Elite), two rings cannot be triggered within the same event. Once the button is pressed, your Doorbell will be focused on operating the recording/ video, audio, notification, chime kit, and powering these events. A motion triggered with a ring (button push) following right after, will also not trigger the ring event/ chime kit right away. It may take several moments after a motion, before the Doorbell is ready to also carry out a ring event, but it is possible to receive both as long as power and network connection is optimal.
For battery powered Doorbell models, two rings still cannot be triggered back to back/ simultaneously, within the same event. A motion followed by a ring (button push), however, can both be triggered for notifications in your Ring app and chime. This will depend on network and mobile device connection variables.
Additionally, if a successful ring occurs during a motion event, it is likely that this event will show as a “ring” in your event history. I hope this helps!