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Airplanes are a type of aircraft (helicopters [直升機], hot air balloons [熱氣球] are also aircraft).
飛機 is a very common Chinese word, but what does it mean?
飛機 - Aeroplane, short for 固定翼飛機 (fixed-wing flying machine)
航空器 - aircraft
飛行器 - both aircraft and spacecraft, i.e. all flying machines which travel through the atmosphere. Includes space rockets.
For reference, spacecraft is 航天器.
Answering the queries directly,
Does 飛機 mean "airplane"? If no, what is "airplane" in Chinese?
Yes, 飛機 means airplane/aeroplane.
Does 飛機 mean "aircraft"? If no, what is "aircraft" in Chinese?
Aircraft is 航空器.
When someone says 飛機, do they mean "airplane", or do they mean "aircraft"?
They should mean aeroplane. If they mean aircraft, well, that's improper use of the language.
飞机 is specifically for airplane
飞行器 is for general aircraft
In English, it has a diiferent words for variety of flying machines. In Chinese, we only use 飞机 as a general term. For example, 直升飞机(直升机), 航天飞机(spacecraft). Airplane is a type of aircraft. Aircraft could be some other machines other than airplane. So, depending on what you are really talking about, aircraft could be interpreted as 飞机(airplane), 飞行器，热气球(hot air balloons) and etc.
飞机 / 飛機 literally means “flying machine”, which is a really inaccurate colloquial term. It can mean an airplane in its narrow sense, and it can also mean an aircraft in its broader sense, and some people even use this word for a space shuttle! Never use this term if you want to describe accurately.
Some well-educated people believe that the broader sense is "wrong", but who cares? All the native Chinese speakers accept the broader sense except some pedants. You can see many pedants on Stack Exchange, but this is not the real world. Welcome to the real world.
If you want to refer to an aircraft, use the term 航空器.
If you want to refer to an airplane, use the term 定翼机 / 定翼機.
If you want to refer to rotorcrafts, use the term 旋翼机 / 旋翼機, which include helicopters, or 直升机 / 直升機 in Chinese.
If you want to refer to a ornithopter, use the term 扑翼机 / 撲翼機.
