I agree: a lot of folks don't think a hypermedia response is an API response, which maybe deserves its own essay.
A hypermedia response is an API response for a hypermedia client, and additional possible API calls to the server are then discovered within that hypermedia response via hypermedia controls (e.g. links and forms) At least, that's my position, but I understand that many people would not call this an API (maybe just a RESTful system?) The language here is pretty darned academic, and Fielding didn't apologize for that:
https://roy.gbiv.com/untangled/2008/specialization
Unfortunately I think that unwillingness to come down a few pegs in abstraction ended up costing a lot in terms of confusion.