No, real precessive action is a result of any force applied to the gyro which results in displacement, or instability. Precession is used to produce useable instrument indications, as in the turn coordinator, and countered to produce a stable gyro in an attitude indicator, with pendulous vanes.
Instrument gyro precession has nothing at all to do with motion of the earth, particularly for an aircraft in flight. Gyro precession takes places as a function of forces out of the plane of rotation, and as discussed previously, occur in the direction of the force, 90 degrees ahead of the applied force, in the same direction as the direction of rotation.
An uneven gyro ruby bearing applies such a force, and the end result in a horizontal gyro such as the heading indicator, is the gyro drifting.
Brand new heading indicators drift including electric ones. This has nothing to do with unscrupulous flight schools renting clapped out airplanes. In aircraft where gyro drift is considered unacceptable, flux gate compasses and other updating systems are used to stabilize the gyro instrument, which is itself serving to stabilize the compass indication.
If you want to get away from gyros that drift, then you have to go a gyroless system. Even advanced INS units and laser gyros drift...even IRS and INS updated gyro systems drift...and these are a little more advanced, and a little more expensive, than what you see in the local flight school. We use triple INS units which most definitely do drift, each updated with independent GPS inputs, plus separate GPS units, and all updated by external ground based navaids as well, where able. This capability isn't available in a rented Cessna...one has a simple gyro on the panel, and it has it's limitations...brand new or well used. The rate at which the gyro precesses and drifts may vary somewhat with age and wear...but it's most definitely a function of precession.
Gyros drift.