An API (as people usually mean by it, anyway…) is really just a simple grammar for the creation and manipulation of objects in the system. Those objects still have types and properties just like any other object, but they tend to be implicit and unreliable. A good DSL is just a more abstract and implicitly verb-oriented API. For example, Lua DSLs can resemble something almost like a sort of typed or tagged DDL. Translating that into an “API” just adds more clumsy legwork to achieve the same outcome.
Agreed - I was tempted to make the same claim that it is still a grammar, but decided against it as I suspected it'd get in the way of making the point. You're 100% right you can write a grammar for an API as well.
In languages where the grammar is sufficiently flexible, the distinction all but disappears, but even in languages where the grammar is rigid and API's stand out like a sore thumb, the API itself still creates a new rule-set that you need to learn.
You can choose to not call that a new language all you want, but the cognitive load is still there.