Yes you're pretty much correct. Nostr is the underlying protocol and the apps you download are "clients" which utilize the nostr protocol.

You're right, this is why your data moves with u regardless of client. Your data is linked to your npub/nsec key pair. So if u can provide the nsec credentials, the client knows to display all of YOUR account info.

The one piece you're missing are relays. Relays also sit on the nostr protocol and to keep things simple think of it as a server or database.

When you write a note in your client (amethyst/damas/etc) that data gets sent to the relays you're connected to. That relay then "relays" the info to all the other ppls clients that are also connected. That's how u see other ppls notes.