What is the official FAA definition of the term enroute?
FAA follows the ICAO definition for describing the phases of flight.
FAA is a member of the CAST/ICAO Common Taxonomy Team that formed the Phase of Flight Definitions and Usage Notes in October 2002. From the CAST/ICAO Common Taxonomy Team site:

Federal Aviation Administration (FAA)- FAA has officially adopted the aircraft make model series (ACFT-MMS) and phase of flight taxonomies as an agency-wide data standard for its internal systems.

So, the enroute phase of flight is the same (just repeating from the same document):

Instrument Flight Rules (IFR): From completion of Initial Climb through cruise altitude and completion of controlled descent to the Initial Approach Fix (IAF).
Visual Flight Rules (VFR): From completion of Initial Climb through cruise and controlled descent to the VFR pattern altitude or 1,000 feet above runway elevation, whichever comes first.
This phase of flight includes the following subphases:
• Climb to Cruise: IFR: From completion of Initial Climb to arrival at initial assigned cruise altitude. VFR: From completion of Initial Climb to initial cruise altitude.
• Cruise: Any level flight segment after arrival at initial cruise altitude until the start of descent to the destination.
• Change of Cruise Level: Any climb or descent during cruise after the initial climb to cruise, but before descent to the destination.
• Descent: IFR: Descent from cruise to either Initial Approach Fix (IAF) or VFR pattern entry. VFR: Descent from cruise to the VFR pattern entry or 1,000 feet above the runway elevation, whichever comes first.
• Holding: Execution of a predetermined maneuver (usually an oval racetrack pattern) which keeps the aircraft within a specified airspace while awaiting further clearance. Descent during holding is also covered in this subphase.