One example I can think of: privacy preserving ML models. Company wants to keep their model safe and also perhaps not to force you to run some classification on your phone for battery savings. You want your inference features to be private, because they e.g. contain your location data. You send them encrypted, you get encrypted results back and decrypt them.

It could have been obfuscated, but assuming we have HmE, obfuscation is more tricky to be done right.

Do we change the definition of encryption meaningfully in this process, though? If so, I don't see how really. It's just that out of set of potential encryption algorithms, for this purpose we would pick ones that are fit for it. It wouldn't be AES I guess... But it would still be encryption. Maybe a weaker one, but that weakness could be coincidental. I'm not familiar with any theorems in that space.

EDIT: I see a note that we lose ability to verify integrity of the data with HmE, but I guess it's still not really changing the definitions - just most of encryption used today also provides integrity guarantees as a kinda nice and desirable side effect, but it doesn't change that the encryption that doesn't still is encryption.