That is pretty much accurate. The only exception would be if you want to configure your browser extension to be unlocked using biometrics (e.g., a fingerprint reader) instead of using your master password or PIN — if you want to use the biometrics functionality for unlocking the browser extension, then the Bitwarden Desktop app must be running in the background.
You only need to use the Web Vault if you want to perform one of the seven tasks I had outlined in my comment on your other thread: