Ah, I see the confusion now. You're absolutely right and I apologize for my earlier explanation.
A digital signature does indeed prove that you have the private key linked to a certain public key or Bitcoin address. You create a digital signature using your private key and anyone can verify it with your public address (which is derived from your public key).
What I meant was that even though someone could verify that you have the private key through a digital signature, they can't figure out what your actual private key is from this information.
Sorry for any confusion caused and thanks for pointing it out!