I disagree with the first interpretation. It's not the best choice of question though. In general, people can either fly a plane or they can't. It's a straight "Yes/No" question. If someone said "No, not really" to that question, I might think that they (like me) had perhaps had one flying lesson but not carried on. Even with one flying lesson under my belt, and a vague familiarity with the controls I was using that day, I would simply say "No" in answer to the question.
You're right that the question dictates what "really" means in the response.
1. Are you hungry?
No, not really.
(Not really - not particularly hungry; not enough to eat anything.)
2. Are you really going to Scotland on Tuesday?
No, not really! I was joking. I'm going to the Maldives!
(Not really - that's not the truth.)
Note that for "really" to be used in the response to 2, it has to be used in the question as well, as in the following example.
If I had heard the surprising news that my friend Mark had qualified as a pilot, the following dialogue might take place the next time I saw him:
Me: Can you really fly a plane?
Mark: No, not really! I just told Monica that because she kept saying I'd never done anything impressive.
Me: Ha ha! She fell for that one hook, line and sinker. She's been telling everyone about it!