Not sure of the roots of airport/aerodrome, so can't comment on which is more correct where.
Courtesy Wiki.
Aerodrome.
Derivation
The word was created by analogy with "Hippodrome" (Greek, from hippos, horse, and dromos, race or course), which was a course for horse racing and chariot racing in ancient times.
General usage
The term is used in International Civil Aviation Organization (ICAO) documents, for example in the Annex to the ICAO Convention about aerodromes, their physical characteristics and their operation. The term "Airport" is also used in the aviation industry. There is not a clear difference in meaning between the two terms.
UK usage
In the UK, the term was used by the Royal Air Force in the two World Wars as it had the advantage that their French allies, on whose soil they were based and with whom they co-operated, used the equivalent term (aérodrome). However, the term "Airfield" or "Airport" has mostly superseded Aerodrome.
United States usage
In the United States, the word was modified into airdrome but has become obsolete since the World War II, though it is still used to describe airplane museums such as Old Rhinebeck Aerodrome, located in Rhinebeck, New York, where dogfights and other flying stunts are reenacted.
Australian/Canadian usage
In Canada and Australia usage it is a legal term of art for any area of land or water used for aircraft operation, regardless of facilities.
The Canadian act says "...for the most part, all of Canada can be an aerodrome", however there are also "registered aerodromes" and "certified airports". To become a registered aerodrome the operator must maintain certain standards and keep the Minister of Transport (Canada) informed of any changes. To be certified as an airport the aerodrome, which usually supports commercial operations, must meet safety standards.
The Canadian government publishes a directory of Canadian Water Aerodromes in the Water Aerodrome Supplement (WAS).