I have generally found that you get credit for the number of flight coupons you have.
Therefore make sure you have 4 coupons ord-nrt, nrt-sin, sin-nrt, nrt-ord. If you have one flight coupon that shows ord-sin and then get a boarding passs at ORD, along with a "re-boarding" pass for NRT--I'm afraid you'll get credit for the non-stop mileage.

Good thing either way is that for ORD-NRT and then NRT-SIN you'll get 9.602 miles while the nonstop number is only 241 less miles at 9,361.