Remember you're crossing the international date line! When you fly US-Japan, you ADD 24 time zones at the date line then resume the MINUS process. Japan is actually 14 hours AHEAD of Chicago (DST) (the same thing as 10 hours behind, but a day ahead).

Using your examples:

12-hour westbound flight:
Departure time: ORD 1pm = KIX 3am (next day)
Arrival time: ORD 1 am (next day) = KIX 3pm (next day)

11-hour eastbound flight:
Departure time: KIX 4pm = ORD 2am (same day)
Arrival time: KIX 3am (next day) = ORD 1pm (same day)

However, it FEELS LIKE the same day westbound because the sun never sets, and it FEELS LIKE the next day eastbound because the sun sets and rises.