Your notes, transcripts and summaries live in your account and sync to every device you sign in on. That is the backup. Audio is deleted once the note is written on every device so there is no audio to back up and nothing to lose with a phone.
What "Back up notes now" does
The button in Settings pushes your notes to your account by hand. You only need it after working offline for a while, when a few notes were written on the phone before it had a connection.
A new phone
Sign in and your notes are there. Two things do not follow: your voice samples, which stay on the phone that recorded them and your private transcripts key, which you link from another device or restore with your recovery code. See Private transcripts.