In other words, these expressions (AGL, AMSL, HAE, AAT) indicate where the "zero level" or "reference altitude" – the vertical datum – is located.
While the use of a barometric altimeter setting that provides a zero reading on the ground of the airport is a reference available to pilots, in commercial aviation it is a country-specific procedure that is not often used (it is used, e.g., in Russia, and a few other countries[which?]).
Most countries (Far East, North and South America, all of Europe, Africa, Australia) use the airport's AMSL (above mean sea level) elevation as a reference.
Rather, it is HAAT (the height above the average terrain (in the surrounding area)) which is used to determine how far a broadcast station (or any other sort of VHF or higher radio-frequency) transmission will travel.
In this case, height AGL is the only important measurement for aviation authorities, which require that some tall towers have proper aircraft warning paint and lights to avoid collisions.