The small piston in the cylinder in the choke assembly sounds like the choke pull-off for a Holley electric choke. It is there so that if you drive the car while the choke is still active, and if you open the throttle a medium or large amount, this pull-off mechanism will open the choke plate to let more air in.... i.e. it 'pulls off' the choke for that situation.
It should be activated by a vacuum port above the primary throttle plate, similar to the ported vacuum going to the distributor advance. (You can see the tiny port on the back of the electric choke assembly at 1:18 into this video; look at the small hole surrounded by brass.)
So smoke going from the carb body into the choke pull-off area should be OK but there should be some seal between choke and carb body so the pull-off can actually work.
Why this seems to be effecting the engine's operation is a mystery to me. This should go to a small port above the throttle plates, and if the choke is removed and exposed that small port, then that should dump some extra air into the carb above the throttle plates. It would normally only effect operation if the choke is closed too much. Is the choke plate opened or partly closed when you have the problem? And if the choke mechanism is now removed and the choke plate is flapping loose, is it closed or open when you get better engine operation?
Some general question on your choke setting and operation:
What temperature is it at your location when you start the engine cold?
Are you pushing the the throttle once to the floor before starting a cold to set the choke?
If you do this, where does the choke plate close to?
And is the choke opening after a few minutes?
One note on the secondary stop screws: the t-port exposure being a square hole is a good starting point, but you should be ready to adjust it from there. Having to close the secondary throttles some more for that stock 318 engine would not surprise me. It is a smaller engine and the stock cam does not need to pull in as much air, so having the secondary plates open a bit may dump too much air into the intake manifold. As an example, with a 340 and a moderate cam, we still had to close the secondary plates so that the exposed slot was smaller than a square by adjusting the stop screw.
And one final note: Does you PCV valve work properly? Remove it so that the end of the PCV valve that goes into the valve cover is exposed but the PCV valve is still hooked to the hose going to the base of the carb. Now listen to the open end of the PCV valve at idle; you should hear very little air flow into the PCV valve at idle. If you hear a loud whoooosh of air into the PCV valveat idle then it is stuck open and letting a lot of extra air into the intake manifold. Now put your finger on the open end of the PCV valve; the idle should not change or change very little. If it does change a lot, then the PCV valve is not good.
Do you have a vacuum gauge? If not, then you should get one and use it to see tell us the intake manifold vacuum at idle.
