The definition of fog is when visual range is reduced to less than 1000 meters by the presence of airborne suspended water droplets in the direction of lowest visibility as viewed by the observer. If the lowest visible range is greater than 1000 meters due to etc. etc. then it is termed mist.
The droplets can be any temperature however if the droplets are observed to freeze onto a structure then the current weather is freezing fog. A dry bulb temperature below zero deg C is usually sufficient grounds to declare freezing fog. I could make a bad joke about hoar frost but that's for another day.
Freezing rain is extremely nasty as water droplets (either supercooled or > zero deg C) coming into contact with the ground or a structure at a temperature below zero will freeze on impact quickly covering the runway / taxiway / road home with an instant sheet of ice.
One surprising aspect of overnight radiation fog is it often thickens shortly after sunrise as a bit of thermal mixing stirs up the cold and moist layers.
Lovely stuff meteorology but the pay is absolutely
