Firstly, mobile first and responsive don't present an either / or choice.
Mobile first is a methodology that you use to approach your design, where you constrain yourself to the limits of mobile. After working that out, it is much easier to modify that design for pc - usually with a better, more focused end product.
Responsive design is a behaviour whereby your layout changes depending on some properties of the device viewing it.
There is nothing stopping a design which was created with a mobile first methodology from being a responsive design at all. In fact I would often recommend that it is.
In terms of recognising a mobile first design, there is no clear way. It is just a methodology for designing that many people find helpful. Usually a mobile first design is more focused and clear on what it is achieving, but you get poor examples of mobile first and good examples of pc first design. A lot depends on the team designing it in the first place.
Mobile first comparison:
Responsive Design illustration: