I cannot attest to what the FAA considers "flight" per the title question, but I believe that their jurisdiction (as well as the means and level of control they exert on aircraft) is less dependent on a legalistic or technical definition of this term, and more about categories and classes of what it considers “aircraft” as well as where such craft actually operate. Let's consider the two examples you offered:
Physically disconnected from the earth as they ride on a cushion air, they are technically "flying". However, they are not classified as aircraft by the FAA and the physical space they operate in is the same realm as boats, cars, trains, and other surface vehicles. The FAA would have no more interest or need to regulate their activities than they would other modes of transportation that operate outside the National Airspace System.
While "air" taxiing, helicopters are also flying per the technical definition. However, air taxi is necessary for helicopters equipped with skids because they don't have rolling wheels to maintain contact with the ground as they move along the surface of an airfield. Since FAA controllers at a tower controlled airport wouldn't necessarily want them to simply takeoff and depart from ramp parking, some means is needed to needed to allow them to move to the runway or other designated takeoff and landing area. As they move along a designated taxiway mere feet from the ground it makes more sense for this movement to be monitored and controlled by the same person controlling fixed wing aircraft, despite the fact that they are actually "flying". (i.e. they aren't talking to a ground controller because they are not considered to be flying, it is because they are on a taxiway!) There is no logical reason to talk to a tower controller until they are ready to enter or depart the tower controlled air traffic pattern.
In summary, while the FAA recognizes rotorcraft as a category and class of aircraft in CFR 61.5, it does not consider hovercraft to be aircraft for regulatory purposes. While you would be free to log your experience driving a hovercraft it would not count as flight time in the eyes of the FAA.