Aerodrome is defined in the ANO and ICAO paperwork, I can't remember the references but google will get them for you.
Airfield comes from the time before runways when they were literally a field from which aircraft took off and landed, early taildraggers always into wind.
Airstrip is usually interpreted as long thin airfield
Airport implies a potential port of entry with some formal facilities for passenger management however basic and compliant with CAA rules for airports.