I'm sorry but I can't follow such descriptions without having a diagram in front of us. The reason I explained the portion of vacuum I did was that it doesn't matter how it gets there only that it exists at the same time the air pump works. That is what you need to test. Tee a gauge to the check valve and verify that the valve gets vacuum when the air pump works cold. Then bypass the circuit to the air pump clutch when warm and see if the O2 sensor voltage drops when the pump is turned on. This won't work when cold because the O2 sensor will be inactive when cold.
Bosch Master, ASE Master, L1
33 years MB technician