"The term API seems to appear for the first time in the article of Ira W. Cotton, Data structures and techniques for remote computer graphics, published in 1968. "
Looong before the advent of web programming. I'd expect a more in depth discussion of API's to at least mention lower level API's like WinAPI, POSIX etc, and how higher level API's are usually created to wrap, or to aggregate existing API's.
After all, the vast majority of SOAP and XML-RPC API's are leveraging those lower level API's to begin with.
It's just an agreed on set of rules about how we're going to exchange data (aka, compute something).