On the next four pictures I show what I took to be the same cistern. I am beginning to think I may actually have visited (for those 4) the smaller cistern indicated as a blue space under the R of Argyroprateia in this map. It fits perfectly with my memories, and the position of the Philoxenos cistern indeed is somewhat to the right.

Detail from an excellent map of Byzantine Constantinople in the Wikimedia Commons by Cplakidas (own work).