There is a rather subjective scale to show several degrees of air frost severity:
Air frost is not necessary to get ground frost or hoar frost; they can form even if air temperature is above freezing point if the surfaces have been chilled by heat emission during a cold night.
In English language, these terms are sometimes used in a confusing way. For example, "frost" is used sometimes to indicate ice which forms on ground during cold nights but other times to indicate air temperature below freezing point which should be called "air frost". "Hoar frost" should indicate big ice crystals forming on very cold surfaces, but it is sometimes used for every frosted surfaces.