Ammonia (NH3) is a gas, which is held in soil as ammonium.
Plant available ammonium nitrogen is the most correct expression, as ammonia has to be fixed as ammonium first.
Ammonia (NH3), which also exists as a gas, forms NH4+ ammonium cations when dissolved in water. The solubility of ammonia gas in water is limited (because it behaves like gas bubbles) and depends on the pH. In acidic solutions, all ammonia gas is easily converted into ammonium. This is why we recommend sampling with containers preserved with sulphuric acid.
Ammonium is what is actually analysed and reported as ammonia - N (nitrogen), as all the ammonia (gas) present in a water sample is converted into ammonium upon acid preservation.
The final result is reported as N (Nitrogen), which is the same whether it is ammonia or ammonium.