Who does the transcription?
When you use Hey Siri to log into Murmr, Siri does the transcription, not Murmr. Your voice is processed by Apple’s Siri speech-to-text. Murmr never receives the audio; it only receives the text Siri sends, then refines it on-device and saves it to your history.
When you record inside the Murmr app (or use “Record with Murmr” and then speak in the app), Murmr does the transcription on-device with Apple’s Speech framework. So: Siri path = Siri transcribes; in-app path = Murmr transcribes.