The air injection code is usually set during a test made with the engine hot. It is this way because the way it tells it tells whether the air is induced is by the O2 sensor voltage dropping below 40mv.
The vacuum necessary in the air inj is that which goes to the check valve that is between the pump and the exhaust. this check valve shuts off the channel to prevent air from being aspirated into the exhaust. Added air is a no-no to a warm O2 sensor system.
The vacuum will go to a switchover valve that turns the vacuum on at the same time the pump comes on. That check valve must have vacuum and the it must open when the pump works and the O2 sensor voltage should quickly drop to 40mv if one were to jumper the air pump clutch.
Bosch Master, ASE Master, L1
33 years MB technician