Just wanted to emphasis about being able to have a local vault in the phone/computer to be able to access passwords offline.

The vault should be super heavily encrypted in the local unit and as soon as internet is available synced with your servers. This local vault might be an option and not default so people can store it locally if needed but keep it safe with you on your servers.
I just know when traveling the world I often end up without internet and still need something stored in the password manager like a note, card number, membership and other things I like to keeps in a safe place.

One could select parts that would be stored in the local vault. Like if I want my notes, Membership info to be available offline I could choose that in the app settings. The rest would be safely stored online safe on proton servers.