The idea is that you can not fasten your seatbelt until you have completed the act of "sitting", at which time you are "seated" (an adjective that describes your current body position) and you should then fasten your seatbelt. While you are "seated" your seatbelt should remain fastened. The verb is "to sit". Sitting is what you "do". The act of "sitting" refers to both moving from any other position into a sitting position, AND remaining in that position (as if you would spring out of the seat if you stopped "sitting"). "seated", in this case, is an adjective. It describes you - the subject. Therefore you can be seated and sitting at the same time. Are you seated? Yes. What are you doing? Sitting. Worry about "seating" later. It generally refers to groups, events, but there are other uses also.

Sorry - 'while (you are) seating' is not correct. Are you perhaps mixing up 'seating' with 'sitting'? You could certainly say 'while sitting'. Seating can refer to 'giving someone a seat', or 'a range of seats' e.g. 'the usher was still seating the large group of people', 'the theatre had several seating options'. This form is possibly used because it is a little more polite.