In this instance amber is your remote signer. So amber holds your key. When a client wants to sign something, it reaches out to amber and asks amber to sign it (you approve)

Your key never leaves amber. That's how it works and why it's much more secure than pasting your name in an app. When you login with nsec, your are trusting the client with your private key. The client uses your nsec to sign things. You have to trust the security model of the client to store your key in a secure way and also trust that they won't just completely rug you to begin with.