As interesting as this article is, it’s overkill. In plain English, an API is an agreed upon way for code to interact with a library of public methods. There is usually a documentation available that includes a list of the methods available, a description of what parameters must be passed to get the desired result, and a description of how to interpret the returned values.
For example, if I create a library of code to access a movie database, I can publish documentation for the interface to that library so other people can access the data. That is an API.