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It's a really good question, because, in order to verify the AD, you have to have the same key you need to decrypt it.

Yep, that's the part that throws me. Is it fair to say that it's a more elegant way to include metadata in the ciphertext, without really messing with the plaintext itself? Ie it's basically "just" a way to distinguish the message from its metadata?

Does that make it like some kind of HMAC?

Yes, in fact, one construction of the AEAD primitive is to use AES-CTR with HMAC to "bolt on" authentication after the fact (AES-CTR on its own is an unauthenticated stream cipher).

You can find an implementation of AES-CTR-HMAC (at a high level where AES-CTR and HMAC are both given) here: https://github.com/tink-crypto/tink-go/blob/main/aead/aesctr...

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