Nearly:
> it was the amount of detail in the content of push notifications
It wasn’t the information in the push notification. This goes via Apple’s server and is a one-bit signal that says ‘there may be some messages waiting for you, you should go and check’ (may be, because Signal sends some spurious push notifications to make traffic correlations harder).
The Signal app then gets the message and asks the local OS notification mechanism to display the notification. If the permissions are set up to display Signal notifications on the lock screen, these are also persisted in a database on iOS (I have no idea why. Is there some way of searching them?). If you’re worried about people with physical access to your device reading your messages, I would suggest that turning off the thing that shows them on the lock screen is probably a good idea.