An API is also the interface used by humans to create programs. When you use a library, you're using its API. This sense of the term API is often lost.
Not sure what you mean. Sometimes the word API is used in one sense, sometimes in another. It's a useful distinction insofar as it allows you to talk about APIs as things used by programmers. I find many developers have a hard time understanding this sense of the word API and as a result fail to apply good API design principles such as SOLID. In fact I think this is often what separates mediocre programmers from good ones.