Can clients of your API talk to other people's APIs using the same interface? If not, then the client is obviously coupled to your server and you definitely do not have a REST application. You have a thick client application. REST is the antithesis of this -- the client is generic and doesn't need to change along with the server.

Putting links in your proprietary data format does not make it hypermedia, it just makes your API easy to reverse engineer. Maybe it also allows you to change your URLs, but it doesn't allow you to change the structure of your data. Hypermedia does. Hypermedia comes in generic media formats that clients know what to do with.

If your API is called "The [company] API" then it is almost certainly not RESTful. If your API is called "[generic type of data] interchange protocol" then it might be RESTful. But we don't usually call that kind of thing an API, we call it a protocol or a format. Really, I don't see how an API, as they are commonly understood, can possibly be RESTful. The main big important point of REST is to not have APIs.