According to what they taught us at school here, the THE would refer to the exact plane the person telling the story is referring to. Exactly THE 180 whe took out to Fairbanks etc.
Although your other suggestions are grammatically sound, they wouldn't match with real life. A pilot won't take just any random plane, he'll take THE plane that is dedicated to him.
Another expression to indicate the certain plane the pilot is flying would be "MY 180 to Fairbanks", but if he doesn't actually own it, "THE" contains all the information needed: The pilot is flying a certain, unique, identifiable plane which has been documented to be under his command at the given time of the event he is telling about. Damn these English sentences need extra reading when drunk tired