Not coil vibration. Different position of the discs relative to the wire, and relative to each other. For example, I expect two tilted rings of coils (relative to the wire) to measure different than 2 perfectly perpendicular and centered coils on the wire. Because of this, and because the 7 discs are close to each other, I also expect a curved/undulated wire (with perfectly perpendicular discs) to measure differently then when the wire is shaped like a straight line.
Again, ideal-coil's voltage is NOT proportional with the current, but with the
speed of the variation of the current. Which means the 7 discs are not a sensor for current alone, but a sensor for current AND frequency. If you measure, for example 1V for 1A at 50Hz, same coil will measure 2V for 1A when the frequency is 100Hz. This is for an ideal coil, without resonance (which resonance can easily happen in practice, but we will put this aside for now).
If you want something proportional with the current alone, then your sensor has to be a resistor. You already have an 8ohm rheostat. The voltage on the rheostat will indicate the current alone (does not depend on frequency).
Again, you didn't say the numbers. How big are those variations you want to fix? What do you expect vs what values you measure in practice? Sorry, can't help you any more by just blindly guessing. Good luck with the experiment.