Frost uses the UK Met Office frost definitions, applied to the forecast minimum air temperature: Ground-frost risk when the low is between 0 and 4 °C (surfaces can still cool below the air on a clear, calm night), Air frost when the air itself reaches 0 °C or below, and we call the coldest nights — around −2 °C and below — Hard frost.
Fog uses the standard WMO and Met Office fog definitions, applied to the modelled visibility: Mist from 1 to 5 km, Fog below 1 km, and Dense fog below 200 m. The number is the model’s visibility for the area, not a reading on your road.