For messages sent through an insecure channel, a properly implemented digital signature gives the receiver reason to believe the message was sent by the claimed sender. While using a digital signature

Encrypting the digest with the sender's public key would be cryptographically useless for authentication because anyone possesses the public key and could therefore forge the signature.

The receiver's private key is used for decrypting messages sent to the receiver, not for verifying a sender's digital signature - using it here would mix up confidentiality and authentication operations.

Encrypting the message digest with the receiver's public key describes asymmetric message encryption for confidentiality, not a digital signature - it would mean only the receiver could decrypt it but would not prove who sent it.

A digital signature is created by the sender hashing the message to produce a digest and then encrypting that digest with the sender's private key. Because the private key is known only to the sender, only they could have produced that encrypted digest, which provides non-repudiation and authentication. The receiver (or anyone) can verify the signature by decrypting the digest with the sender's widely available public key and comparing it to a freshly computed hash of the received message.