[Benanov]

Correct.

With public/private key pairs, encrypting anything with the private key means that you use the public key to decrypt that same thing. This means anyone (as the key is public!) can decrypt the thing. So if you get the public key, and if the thing decrypts successfully, then you know that the corresponding private key was used to encrypt the thing. This is considered proof that the private key holder encrypted the thing / sent the message, and that's why everyone calls it "signing" instead of "encryption" - you send the cleartext thing along with the encrypted thing.

For private messages, you encrypt with someone's public key and have them decrypt with their private key. You'd sign it with your key, and that person would verify the signature with your key. That's 4 keys you need to worry about.

This doesn't even begin to consider key rotation, perfect forward secrecy, multiple recipients, etc.