I don't think specific gravity matters.
Consider the drawing. When the ignition switch is in the run position, if nothing is broken, there is always a current flow through the fuse, then the relay coil in parallel with the resistor wire, then through the thermistor, and then to ground.
When the thermistor is submerged in fluid, whether water or gasoline, enough heat is transferred away from the thermistor to keep the thermistor resistance low enough, resulting in the relay being energized. There is enough voltage drop across the relay coil to pull it in and turn out the indicator light.
This is why the resistor wire is important, it is sized so that the "voltage divider" circuit works properly.
When the thermistor is not submerged in fluid, there is less heat transfer, so the thermistor heats up. It is this repeatable property that is fundamental to success in this design. When the thermistor heats up, the resistance increases. As more voltage drop shifts from the relay coil to the thermistor, eventually the relay does not have enough potential and it drops out, and the indicator light comes on.
There is a time component to this, and as stated previously I would guess it is in the 10's of seconds, not minutes. This allows for variability in relay coil, resistor wire, and thermistor manufacturing. It also prevents the light from flipping on and off when the level is close to the low level and you hit the brakes causing sloshing.
For the OP, I would do these things next:
1. Figure out how to ground the wire 29 with the thermistor plugged in. I understand that you would want to do this in a non-destructive manner. Based on what you wrote, if you ground 29 the light will go on, and if you lift the ground (open circuit) the light will go out. Is that the case? If so everything is in place, but now we need to determine if the problem is in the thermistor or with a bad quality ground.
2. With the same setup, a temporary known good ground on 29, get the circuit hot again, with the thermistor in the air. No need for a bucket until we can get the light on. If we do, submerging in water should turn it off after an unknown (yet) time delay. I would really be tempted to use thermography to measure the surface temperature of the thermistor. If you have a quality meter, you could try to get in series and measure amperage. If this works (I would probably wait at least 5 minutes just because I don't know how long it takes to heat up), then submerge and wait again to see it clear. It this all works, then your wire 29 is bad and you need to chase that. If it does not work, the thermistor is bad.
I'll try to get out and check mine this weekend.