Flightcrew oxygen is protective and can be selected to suit various circumstances.
Passenger oxygen is a supplemental arrangement intended to cover the depressurisation case and provide a bit extra partial pressure of oxygen during the resulting descent to lower levels. As such the system utilises rebreathing. The comparatively low oxygen flow rate is supplemented by cabin air to provide an oxygen enriched mixture but certainly not straight oxygen.
The bag provides the means to introduce cabin air into the breathing activity and a mixing volume for the cabin air, supplemental oxygen, and exhaled lung air. The extent to which the bag does/doesn't inflate/deflate during use is not a critical consideration .. however, by gripping the bag, one can force a small degree of inflation to check that the oxygen supply actually is functioning.
The cabin air component is the reason that the cabin oxygen system is not a good idea in a smoke filled environment .. the passengers just get a lung full of smoke ....