I was taught the following in yeshiva:
The difference between emunah and bitachon is as follows: emunah is that you believe in a specific thing that you want to believe in. For example, if I buy a lotto ticket and really believe that Hashem will make me win, that is emunah. I think this is what will happen, and really believe in it.
Bitachon is belief that even if I don’t win, that is what was best for me – I’m better off not having won!
For the airplane example, I see it like this. If you thought a plane would crash, would you get on it? If you really believe it won’t crash, that’s emunah. If you say it doesn’t matter, because Hashem will do whatever is best for me, and if it crashes, that’s a good thing for me, then that’s bitachon.