I found this after a bit of research. It even lists REC and REC 2 as part of this relatively new genre within horror films.
Here is the definition:
Found footage is a genre of horror filmmaking in which all or a substantial part of a film is presented as an edit of recovered footage, often left behind by missing or dead protagonists. The events onscreen are seen through the camera of one or more of the characters involved, who often speaks offscreen. Filming is often done by the actors themselves as they recite their lines, and shaky camerawork is also often used to maintain realism.