A digital signature:

A digital signature is created by hashing a message and encrypting that hash with the sender's private key. Any recipient can verify authenticity - confirming the message came from the claimed sender because only the private key holder could have produced the valid signature - and integrity, because any modification to the message after signing will cause the hash comparison to fail upon verification.

Automatically exchanging shared keys describes the Diffie-Hellman key exchange protocol or similar mechanisms, which is a separate cryptographic function from signing.

Decrypting data to its original form describes the decryption operation performed with a symmetric or asymmetric key, not the purpose of a digital signature.

Providing a secure key exchange mechanism over the Internet describes protocols such as Diffie-Hellman or IKE used in IPsec/TLS, not the role of a digital signature.