In the image below, what dictates the order of the languages on the top banner (e.g. why English and then French?
The rules don't provide any directive on order of languages. It seems likely that the order is "Alphabetical in English".
So Arabic (the only right-to-left language is placed first (on the right). Then the left-to-right languages are in alphabetical order Chinese, English, French, Russian, Spanish, from left-to-right
The rules of procedure of the security council name these as the official and working languages in alphabetical order.