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They do both generate lift in the same way, the collective lift generated by the blades spinning creates a net upward lift and a moment that would cause the helicopter to spin, the blades at the tail create an opposing moment to stabilize the helicopter. Don’t think about it in terms of the air being directed downwards but the helicopter being pulled upwards by the low pressure area above each blade.
I don’t understand what you mean by a rotor with blades parallel to the ground creating no lift. This would only be true if the blades had no camber or twist, which isn’t the case for any helicopter. Also, I think you had a typo, you said
>The collective changes the pitch angle
That doesn’t make any sense to me. By the way you are right that they generate lift in the same way a variable pitch propeller generates thrust, but that **is** the same way a wing generates lift, the lift force is just directed forwards for the propeller rather than up for the wing.