The fuel gauge in effect measures the resistance on the line to the sensor in the tank. When the tank is full there is very little resistance but as the fuel level lowers the resistance is increased. The fuel gauges are notoriously susceptible to a bad earth between the sensor in the tank and the negative pole of the battery. Naturally this normally passes through the chassis and it may be that you have a bad connection somewhere. I was told at Caerbont Automotive Instruments (Smiths) that they had a lot of problems with the London taxis and in the end recommended a direct earth line from the earth on the sensor to the battery.
IMO, a bad earth is simply ... a bad earth! i.e. it would give a constantly false reading, regardless of whether engine was on or off. So it doesn’t fit your symptoms. Your own idea of variable voltage due to alternator running sounds more feasible BUT the supply to instruments is usually connected through a voltage stabiliser to avoid that very issue. Nowadays, stabilisation is usually part of vehicle’s electronics rather than a separately identifiable component, so if faulty, may be difficult or costly to track down.
One other consideration (& a more likely scenario) is that fuel sensors are usually damped in some way, otherwise every time you accelerate or slow down, the gauge would be very erratic as the fuel sloshed about in the tank. Are you being too quick to test the effects of engine running or not. Try letting everything settle for a good few minutes then try looking with engine running or not, without moving the car.
