The FAA definition is in 14 CFR 135.100 and the pilot/controller glossary (quoted here):
TAXI- The movement of an airplane under its own power on the surface
of an airport (14 CFR Section 135.100 [Note]). Also, it describes the
surface movement of helicopters equipped with wheels.
Interestingly, "pushback" is not in the glossary at all.
This matches the Wikipedia definition that you mentioned, although not the ICAO one from DeltaLima. I think this definition makes sense, because taxiing doesn't necessarily have to be before a takeoff or after a landing. It's possible that the aircraft is simply taxiing from one point to another on the airport surface, e.g. from the fuel pumps to the hangar. That might be more common in GA, though: when I see airliners repositioning on the ground they're usually being towed.
You also didn't mention air or hover taxiing, but I think it's fairly obvious in that case that the helicopter is under its own power. Here are the FAA definitions, for completeness, both from the P/CG:
AIR TAXI- Used to describe a helicopter/VTOL aircraft movement
conducted above the surface but normally not above 100 feet AGL. The
aircraft may proceed either via hover taxi or flight at speeds more
than 20 knots. The pilot is solely responsible for selecting a safe
airspeed/altitude for the operation being conducted.
HOVER TAXI- Used to describe a helicopter/VTOL aircraft movement
conducted above the surface and in ground effect at airspeeds less
than approximately 20 knots. The actual height may vary, and some
helicopters may require hover taxi above 25 feet AGL to reduce ground
effect turbulence or provide clearance for cargo slingloads.