The vacuum applied to the piston is calibrated to open the choke valve slightly after engine start to prevent an over rich condition. Necessary for start, but not for idling. The spring that heats up or the electric coil that heats up at a set rate open the choke fully at the calibrated time based on engine being fully warmed up. This allowed a car to function properly in cold weather and be driven immediately after start up, while progressively opening the choke valve and changing mixture quality to meet conditions as engine heated up.

For divorced choke carbs the vacuum pull off canister does the same thing as the Choke piston. Other early carter AFBs had a piston integrated into the top to pull off the choke valve via vacuum.

Normally the inlet receives air from the air cleaner through the exhaust manifold to properly determine engine temperature. For electric you could allow air flow from air cleaner since hot air not needed. Or you could plug it. All depends on if that was original housing or an added electric to a thermal housing. Typically there is a metering orifice in the carb body where the choke housing attaches to the carb body. This meters that air leak into the carb which is typically below the throttle blades and an unmixed air inlet. I believe newer electric chokes from EB do not have that air inlet.

Remember older carter AFBs had vent holes to outside air above the throttle blades for venting purpose. They were a calibrated air leak that did not mix fuel via the venturis.