The ICAO designations are: air/ground taxi; and air transit. The former is well understood and the latter is:
"intended to permit the movement of the helicopter above the surface, normally at a height of not above 30m (100ft) above ground level and at ground speeds exceeding 20kts."
"Air transit routes, however, require comparatively large amounts of airspace (widths of up to 200m at night), which must be kept clear of all obstacles as well as corresponding areas of ground below them, which must be suitable and of sufficient bearing strength to permit safe emergency landings."
"Air taxi routes are to be selected so as to permit auto-rotative or one-engine-inoperative landings such that, as a minimum requirement, injury to persons on the ground or water, or damage to property are minimized."
Under the ICAO designation, 'air taxi' and 'hover taxi' are synonymous.
Jim