Why is Rei's name written in Western style, while the other two are in Japanese style?

they are all western style... the others are not japanese to begin with.

I think the question is why Rei is the only one whose surname doesn't come first, as is typical in Japanese and other asian cultures.

Mari and Asuka may have western middle names, which sort of screw with the typical format, but their surnames are where you would expect them for a Japanese person.