This is the second time I've read this Ali Baba paper and I'm pretty sure I got hung up on this, too. And I don't see what that property is good for (though I am aware that other people do).

Maybe the paper should have a preface. It might work for entertaining someone who already knows all about zero knowledge proofs, but it doesn't seem to do much to educate those who do not.

Try any cryptography textbook! This is the whole point of zero knowledge proofs: that you can convince someone, yet they have no evidence (i.e, "zero knowledge") that they can take to another person and expect to convince them.

No, the zero knowledge part is that they only learn that you know a secret, but they don't learn anything about the secret. It's not about convincing others.

If you get some piece of evidence that you can take to others, then you do in fact learn something about the secret.