The books use "precession" as a way of getting stupid students to sort-of understand why the disc behaves the way it does.
The swash plate and links feed in a pitch angle to the blade. Blade's angle of attack goes up, and generates more lift.
Lift is a force. F=mA
The lifting force starts to accelerate the blade upwards, with a rate depending on the mass of the blade. It takes time from when the max input is fed in, with the max rate of acceleration, until the blade reaches its max position.
While this is happening, the blade is rotating about the mast. By the time the blade reaches its max height, the mast has turned ABOUT 90 degrees. (As stated before the R-22 has turned only 72 degrees, mainly because of the light blade.)
In the meantime, the swash plate has been reducing the pitch angle, so the blade is now being told to flap down, and so it goes on around the full circle. The max acceleration up or down happens ABOUT 90 degrees ahead of where the blade reaches its highest or lowest point.
Some helicopters have the lead angle set 45 degrees ahead, but also have the control rods moved 45 degrees around the mast. Others have the control rods running up the side of the mast, so the about-90 degree lead input is easily done.
And don't get into a tizz about "Flapping to Equality" because it only happens if the cyclic is held fixed, and it results in an eventual crash. People think that in forward flight, the advancing blade must flap up to reduce the extra lift from the forward airflow, and the retreating blade flaps down. All you need to do is look at the disc in forward flight, the advancing blade is flapping down, and retreating blade is going up.
As soon as the cyclic is moved to prevent the flap-back, the blades obey the swash plate. Poke it forward, disc goes forward. Simples...