I think the key may be that the rich roommate actually prefers a room, while the poor roommate does not (according to their assigned valuations) have any room preference at all. So if the poor roommate is second, they make the most choices in this calculator and their lack of preference makes the split even, whereas if the rich roommate is second, they make the most choices and their stronger preference causes them to pay more.
Even so, the order still seems to matter somewhat, and I would have expected it not to matter at all.
Even so, the order still seems to matter somewhat, and I would have expected it not to matter at all.