In general, API architecture spans the bigger picture of APIs and can be seen from several perspectives:
API architecture may refer to the architecture of the complete solution, consisting not only of the API itself, but also of an API client such as a mobile app and several other components. API solution architecture explains the components and their relations within the software solution.
API architecture may refer to the technical architecture of the API platform. When building, running and exposing not only one, but several APIs, it becomes clear that certain building blocks of the API, runtime functionality and management functionality for the API need to be used over and over again. An API platform provides an infrastructure for developing, running and managing APIs.
API architecture may refer to the architecture of the API portfolio. The API portfolio contains all APIs of the enterprise and needs to be managed like a product. API portfolio architecture analyzes the functionality of the API and organizes, manages and reuses the APIs.
API architecture may refer to the design decisions for a particular API. This is what we usually refer to as API proxy architecture. It includes architectural design decisions, non-functional design decisions, frontend and backend design decisions. To document these design decisions, API description languages are used.
So, what is the difference between API architecture and API design? API design is one aspect of API architecture. API architecture has a wider scope, considering also the API solution, API platform and API portfolio. In a way, the API architecture defines the frame, in which API design can take place and does make sense.
API design can only be effective if the overall API architecture is already in place. If you are interested in the big picture of API architecture, I recommend the book API Architecture from the API-University Series. If you are interested in the practical details of API design, I recommend the book Restful API Design from the API-University Series.