Does Craft CMS have an API?
No, Craft doesn't have a REST API (edit: or any other web service) built-in. The "API" mentioned in the first quote (and elsewhere related) would be the more generic meaning of the term and refers to the service layer.

With a very consistent MVC based architecture based on Yii, Craft is well suited for REST implementation, though. See this G+ thread for P&amp;T weighing in on the subject.

As a native REST API isn't part of the core, one would have to added to Craft via a plugin. 

There's at least one plugin out there, though I haven't tried it (or any others) myself.

Edit

It should be mentioned that after I wrote the above answer, Pixel and Tonic has released the Element API plugin - making it easy to build a REST API with Craft. Element API is pretty basic (only GET requests), but can probably be extended with a custom plugin to support additional HTTP methods.

P&amp;T has also stated that they want Craft 3 to have a full, built-in REST API at some point.