>> I don't think a digital signature is a Zero-Knowledge Proof because someone else could copy and paste the signature and then it would look like they know the key, and because other third parties could check whether the signature was valid or not.

One of us is confused. You can't copy a digital signature in a useful way. Without the message it doesnt mean anything. With the message its proof that the message was signed by someone with the private key.

To meet your second two (arbitrary) requirements, have the signer encrypt the signed message with your public key before sending it to you.

They're not my arbitrary requirements, see https://en.wikipedia.org/wiki/Zero-knowledge_proof

Specifically:

> In light of the fact that one should be able to generate a proof of some statement only when in possession of certain secret information connected to the statement, the verifier, even after having become convinced of the statement's truth by means of a zero-knowledge proof, should nonetheless remain unable to prove the statement to further third parties.

I think it’s the original quote that is unclear:

> a digital signature proves your possession of a private key without revealing that key.

Signatures do not themselves do this; but they can be used to construct a protocol that does (e.g. the provee provides a random challenge that the prover must sign). But still this is not AFAIU a zero-knowledge proof as the signature is itself “knowledge”.