What are the official definitions of regions in the United States like "the Midwest," "the South," or the “East Coast”?

The U.S. Board on Geographic Names does not define the extents of geographic regions, nor are they recorded in Geographic Names Information System (GNIS). The names and boundaries of regions are application-driven; that is, defined by a specific entity on a case-by-case basis according to a specific need, characteristics, or perception. 

Individuals might agree on the core of a region, but the outer extent will vary. The criteria or application would have to be defined, such as physiographic, ecological, political, cultural, and other applications. 

Regional definitions applied by any organization reflect their particular needs, not a government standard.