Strictly speaking, you can use hypermedia without REST, too. But that's the lineage of the last few years of what people are talking about.
As an example, an api for AirBnB's mobile website might expose a /srp endpoint that can be used by a thin client to render the results recursively. Again, the way I've internalized it, to be a REST api I'd expect a client to request results from a /listings endpoint of some kind.
I think there is a lot of confusion and misunderstanding and that precise language is better for everybody. What should we be calling these things? I'd love your thoughts on it.
I think that the true heart of the hypermedia approach is actually "how do I deal with change over long periods of time in a scalable fashion?" Everything else flows from this. Exposing behaviors instead of data is one facet of the approach.
> What should we be calling these things?
Which 'these' are you referring to?