Missing Notes after Update…
My phone updated 1 night ago and the notes tied to my email account are all gone. None of the other accounts are missing. How do I recover these?
iPhone XR, iOS 16
Where were the Notes? See the pic below. You will get your answer
The various options that you are referring ("the second text menue") to will exist only if your Notes are on iCloud or the "On My Phone" Folder.
The 2 pics below are self-explanatory.
If you're missing notes on your iPhone, iPad, or iPod touch
Notes are always synced through Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other email Account
Any data on your device that isn't synced regularly will be stored in your iCloud Backup. With Backup turned on, iCloud takes periodic snapshots of this data to make sure it's easy to restore any time you get a new device. Between syncing and backing up, all of your data is kept safe in the cloud.
What does iCloud back up? - Apple Support (IN)