Programming, and the "A" is for application, and the application is a user. Albeit, one that is not human, but programmed. So the programmer is programming the application user to behave in a way that can interface with the application exposing the API.

Hence, and API is a UI. When I write user stories, all external applications are users (e.g., Payment Processor User), and the internal application is the system user ("As the System, I would like to...").