I don't believe they are confused about the nomenclature, as everything the parent said was correct. In terms of there not being standard ways to add an HMAC to

I don't believe they are confused about the nomenclature, as everything the parent said was correct. In terms of there not being standard ways to add an HMAC to an encrypted string, there are plenty of standard ways to do that. How do you think people implemented secure channels before Cisco released AES-GCM to speed up their VPNs? Just let attackers modify payloads? SSL/TLS, SSH, IPsec and many other widely used secure channel protocols were over a decade old before anyone standardized an authenticated encryption mode. But none of these widely used and well-standardized protocols require an authenticated encryption mode even though they all have both authentication and encryption as security properties. Authenticated encryption doesn't give you anything other than performance over the standard encrypt + tag approach (which is still widely used). And AES-GCM has some drawbacks that encrypt + tag do not (although the performance gains usually trump the drawbacks).