You are not missing anything. There is no meaningful distinction in cryptography between ‘integrity’ and ‘authentication’ of data on a channel between two parties. Any attempts at distinction in the context of an adversary wallow in confusion of epistemology without a difference. (If you really like the cute acronym CIA for spook stuff, maybe use the A for ‘availability’ instead!)
The words are not entirely synonymous, of course—for example, a dishonest politician does not lack ‘authentication’, and logging into your bank's web site is not doing ‘integrity’—but it sounds like you're asking how they apply a protocol in which there are three things:
- A sender; call her Alice.
- A channel or medium for exchanging messages, such as the internet, or a disk for storing and retrieving files.
- A receiver; call him Bob.
Alice's goal is to transmit a message to Bob using the channel. Presumably, the channel is subject to malicious tampering by an adversary; otherwise there's no security question here. Presumably, Alice and Bob know something about one another a priori; otherwise when Bob receives a message there is necessarily no way he can distinguish a message Alice sent from a message the adversary sent!
Under what circumstances, then, does the protocol have authentication or integrity? Either way, what it means is that—whatever steps Alice takes to send a message, and however Bob verifies the authenticity of a message—the probability that an adversary succeeds in fooling Bob into accepting a forgery is negligible.
In cryptography, we formalize the notion of ‘authentication’ or ‘integrity’ with the EUF-CMA game—existential unforgeability under chosen-message attack. In the EUF-CMA game, Mallory, our heroic adversary, can ask Alice to send any message of Mallory's choice, in order to study how Alice is authenticating messages. Mallory wins the game if they can fool Bob into accepting any message Alice didn't send. Mallory would win the game even if the message is totally meaningless to Bob. The protocol is said to provide EUF-CMA security if the probability that Mallory wins this game is negligible no matter how clever they are.
The EUF-CMA criterion applies to both symmetric-key authenticators, also known as message authentication codes, and public-key signatures; the main difference between authenticators and signatures is whether the power to send a message is the same as the power to verify a message, or whether the powers are asymmetric so that when Alice signs a message, anyone—Bob or any third party—can verify it too.
Outside cryptography, one might reasonably consider a sort of unintelligent adversary—a channel with random noise—and one might use an error-detecting or error-correcting code to provide integrity, meaning that the probability of undetected corruption is substantially smaller with the code, whereas the term authentication doesn't really come up in coding theory at all, so there's no contrast to make between the terms in that context. One might even use a ‘cryptographic hash function’ like SHA-256, truncated if you have limited bandwidth, but one would be better off with a CRC.