Unfortunately, one can't try to make sense of idiomatic expressions (why do we say "raining cats and dogs" anyway?), particularly for a language as haphazard as English.
We can be "on" or "in" an airplane, but the ordinary usage is "on." Likewise, we also say that we are "on" a bus or "on" a train, meaning we are traveling by means of it en masse. But, say there is a hostage situation, and the plane is sitting on the tarmac while the hijackers decide what their demands are. The chief of security would probably want to know how many people are "in" the airplane. The passengers would also be "on" it in that case, but "in" becomes possible when you look at the airplane as a container and not a vehicle.
However, if you say that you are "on" the car, the meaning becomes that you are sitting on the roof or somewhere on top of the car...
As for "at," it is generally used to designate a specific location that is not a vehicle and not a city or country (use "in" for cities and countries), such as at school, at work, at church. However, sometimes you can replace "at" with "in," but this also sometimes changes the meaning. "At school" means you're physically located on the grounds of the school, whereas "in school" means primarily that you are a student and enrolled to take classes. On the other hand, saying that you're "at the library" or "in the library" means approximately the same thing, although "in the library" emphasizes the fact that you're in the interior of the library building.
There is not always rhyme or reason to English prepositions, and the best way is to learn them in the context in which they are used.