How is the following different than RSA Signing of a document? If the goal is to demonstrate that you are who you say you are, I think the following will work: Bob wants to know that you are Amy. Bob encrypts a number using Amy's public key and then asks her what that number is. If Bob gets back the number, "Amy" at least had Amy's private key.

From what I have read, this is not what signing a document using RSA is. It is clear that it requires more emails to be exchanged; but is the fundamental idea different from Bob encrypting using Amy's public key and wanting to get back the correct decrypted message?
Two scenarios:


Alice sends a document to Bob that she has signed using her private key. Bob verifies the signature.
Alice sends a document to Bob. Bob encrypts a random message for Alice's public key. Alice decrypts it and sends it back to Bob. Bob verifies it's the same.


Both will only be possible if Alice has access to the private key, so you might think they are similar. However, the latter does not make any connection between the document and the authentication attempt. In fact, it allows a man-in-the-middle attack:


Eve intercepts a document meant for Bob, so Alice expects an encrypted message for decryption.
Eve sends another document to Bob, claiming to be Alice.
Bob encrypts a random message using Alice's public key and sends it to Eve.
Eve forwards it to Alice, who decrypts it and sends it back to Eve.
Eve can now send the decrypted message to Bob, convincing him she is Alice and that the document came from Alice.


In comparison, with signatures the document itself is signed. The signature cannot be transferred to a different document, or it will no longer verify. Moving it would require either breaking the asymmetric system to find the key or finding a hash collision.