Well, libraries are collections of APIs and sdks are usually collections of libraries. So they’re unfortunately kind of interchangeable when discussing them.
An API is a specification of what functions are called and how they behave. See for example “Microsoft Windows provides the Win32 API” and “WINE provides the Win32 API on Linux” and also “Photoshop provides an API to write plugins” and “Affinity Photo provides the Photoshop API to support Photoshop plugins”.
When people, who don’t even know that finished the Portal64 ROM uses original Portal PC art assets copyrighted by Valve, try to lecture me about Valve acting as a henchman for Nintendo because of Nintendo APIs, I obviously dismiss them because they clearly have no clue about anything, even if by pure luck they may have a point regarding your definition of API use.