The dominating force on the blade is centrifugal, by huge margin, so it really always moves in a plane, with the changes in lift just tilting that plane relatively slowly to the rotation.
The reason for flapping hinges or flex element or teetering is to allow the fuselage to follow the rotor tilt at its own pace without overstressing the mount.
Now both effects you mention are true at the same time, and they cancel each other out. As you push the cyclic forward, lift is increased on the retreating side, which tilts the rotor forward (with the fuselage following) and the helicopter accelerates. That increases the lift on the advancing side, and that in turn stops the forward pitching motion.
This is good for controllability, because it means the helicopter will settle on some forward speed corresponding to the displacement of the cyclic control rather than keep pitching forward.
The ‘flapping’ term is perhaps a bit misleading. It's better to view the rotor as the primary component, a gyroscope that maintains its orientation like any other free gyroscope due to conservation of angular momentum, and any changes in lift, whether caused by cyclic input or by speed differential between the sides, cause it to precess.
The fuselage is just hanging underneath. In some helicopters there is some torque carried by the hub, but in teetering rotor helicopters, and that's common design in light helicopters, it's literally just hanging there like a pendulum.
Now the swashplate is designed so that when the rotor is not in line with it, the blade pitch is manipulated so that the rotor tries to re-align itself with the swashplate. This design ensures that the rotor won't continue to precess until it hits the helicopter even when there is no torque transferred by the hub.
Except you have to add the effect of differential lift, and that means during forward flight, the rotor will actually spin tilted somewhat aft relative to the swashplate, with the difference increasing with forward speed.