According to the information that @aeroalias found - which I'm sure is correct - the ICAO definition is used "internally" by the FAA:
FAA has officially adopted the [...] phase of flight taxonomies
as an agency-wide data standard for its internal systems.
However, as far as I can see, that hasn't resulted in a clear definition in the regulations or the FAA's publications. There's nothing in 14 CFR 1.1 or the PCG and the best I could find - apart from the definition you quoted from the IPH - are sources that imply that en route (or "enroute") refers to cruise flight only and doesn't include departures and approaches. That would conflict with the ICAO definition.
First, section 2 of the IPH says what en route airspace is:
The en route airspace structure of the National Airspace System (NAS)
consists of three strata. The first stratum low altitude airways in
the United States [...] are called Victor Airways. [...] The second
stratum high altitude airways [...] are called Jet Routes. [...] The
third stratum allows random operations above flight level (FL) 450.
That's a fairly specific definition that strongly implies that - at least historically, i.e. pre-GPS - en route meant "on the airways".
Second, TERPS doesn't define en route either, but it does repeatedly suggest that approaches and even feeder routes are not en route.
Section 3 on initial approaches says:
In the initial approach, the aircraft has departed the en route phase
of flight and is maneuvering to enter an intermediate segment
Section 2 on feeder routes says:
[...] the angle of intersection between the feeder route course and the en
route structure must not exceed 120 degrees
That seems to me to mean that a) by maneuvering to start an approach you are no longer operating en route, and b) feeder routes are not considered en route.
Finally, if you look into some of the FAA's legal interpretations, they seem to say that enroute means cruise flight:
[...] pilots serving as SIC during the en route cruise portion of the flight
[...] the pilots identified in both of your questions serve exclusively en
route, during the cruise portion of the flight, [...]
I assume that most pilots would agree that descending/maneuvering to start an approach is no longer cruise flight. But unfortunately there's no formal definition of "cruise" that I could find - apart from possibly the ICAO definition - so that isn't entirely clear either.
So my take - and I may be completely wrong - is that even in the absence of a formal definition in the regs, there's quite a lot of FAA material that states or implies that en route means cruise flight on an airway or other defined route, and it doesn't include departures and approaches.
But for a 'real' answer, you might have to ask your local FSDO for their opinion, or even ask the FAA for an interpretation if you need it.