Trier was the primary chancellor for all of these realms which, seem to have been conjoined together under Trier as a sort of Middle Francian-like entity by 1200. By this virtue, that Trier was within Lotharingia, was its chancellor and yet also the chancellor of the realms of Arles and Burgundy, these three-two were all one. So, in my opinion it is best to refer to these lands all as Lotharingia, in respect of the conception of Middle Francia and Trier's role in conjoining these areas together into a demesne outside of Germany, Italy and France yet within the Empire. There is no better term for Middle Francia, than Lotharingia, unless you wish to say something that strains the tongue Lorraine-Burgundy-Arles.