Again, from the 54 service manual: m
The fuel gauge system consists of an electro magnetic unit on the instrument panel and a tank unit. the system is connected to the battery through the ignition switch. The gauge operates only when the ignition switch is on.

The electro- magnetic unit on the instrument panel has two magnetic circuits. the two poles of the electro-magnet are equipped with individual windings which produce two distinct magnetic fields.

One of the windings is connected from the switch terminal to a ground. With the ignition switch turned on, the electro-magnet is energized and tends to pull the armature ( to which gauge hand is attached) towars the "empty" gauge position. This winding is called the constant field.

The other winding is connected from the switch to the tank unit and is grounded. This winding is a variable field winding and it pulls the armature towards the "full" gauge position.

The tank unit consists of a float, a float arm, and a variable resister. The tank unit assembly is mounted on a base which is attached to the tank. It is held in place by a retainer and a rubber seal. A single wire connects the electro-magnet and the tank unit. The tank unit is grounded.
Your 3 wires are a ground wire. a power wire from the switch (connected to both armatures), (marked SW) and a variable field wire (marked GA)

Another note: As the fuel in the tank moves from "full" to "empty", a change in the magnetic field surrounding the variable field magnet takes place. Then the gauge hand moves indicating the amount of fuel in the tank.

When the fuel tank is empty, the float has dropped to the bottom of the tank. Downward motion of the float moves the variable resister contact finger and increases the amount of resistance in the variable field circuit. The increased resistance decreases the current flow and the magnetic pull of the variable field magnet.

The decreased magnetic pull of the variable field magnet allows the constant field magnet to move the gauge hand to the "empty" position..
When fuel is added to the tank, the float is lifted. This moves the variable resistance contact finger, decreasing the resistance in the variable field circuit.

This decrease in resistance allows more current to flow in the variable field circuit, increases the magnetic pull of its magnet and pulls the gauge hand towards the "full" position.

Hopefully you can figure out how to test the wires to be sure they are good. You also want to be sure the sending unit in the tank is good. With the float out of the tank, you should see a change in resistance as the float arm is moved up and down. The bood does not give resistance numbers. The test is how bright a 6 volt buld in a test light is as the arm moves. Remember, there is less sresistance as the float arm is lifted.