It appears Americanization is the root of all confusion. From a dated USAF perspective I'll toss in my two cents.
An airdrome is the runway, taxiways and associated buildings (hangars, control tower and operations buildings). I read that in a glossary somewhere.
An airfield also includes the other support facilities including dormitories, administration offices and other nonaviation facilities.
An airport is something civilians use. It would have to include an airdrome and might even be an airfield.
The term 'helo' is acceptable by helicopter personnel from those limited by writing space, ignorance or mental capacity. The term 'chopper' should only be used by the mentally incapacitated, those ignorant of aviation terminology or veterans suffering from infantry flashbacks of Vietnam.
Airplanes are those powered aircraft that must move forward to develop lift, the one that should endeavor to land prior to ceasing forward motion.
Those that speak English outside of North America (or North of 49 degrees within North America) probably should replace 'air' with 'aero' in everything above.
Someone please point out my errors.