In response to ospreydriver, i trained in the US and now fly in the UK and i believe there is a difference in the meaning of "air taxi" and "hover taxi" between US and UK.
In the US as you say air taxi is normally higher and faster than a hover taxi.
However in the UK, according to CAP 413 the term air taxi is meant to replace hover taxi.
"The term 'AIR TAXI' shall be used when it is necessary for a helicopter to proceed at a slow speed above the surface, normally below 20 knots and in ground effect" (ICAO).
Hope that helps.