Does 飛機 mean "airplane" or "aircraft"?
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.