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ZAPPY:
In the circuit you referenced there are many junction points, different wires, switches, and loads.
The source voltage at the points close to the actual source, ground some place on the chassis and the point at the top close to the power source, is 12.7 V. If the engine was off and you were measuring across the battery terminals, then the voltage might be slightly higher than 12.7 V.
Fundamentally the drawing shows a lot of series connected wires going to two bulbs in parallel.
A note at the bottom says there is little or no voltage drop from the ground side of the bulbs to the chassis. Thus you have a low resistance path in this part of the circuit relative to the load current.
As a frame of reference I will guess that the pair of lamps at normal voltage draw about 5 A. That would make each bulb about 12*2.5 = 30 W. This is a resistance per bulb of 4.8 ohms and in parallel 2.4 ohms. I will make an invalid assumption for simplicity. This is that the bulb resistance will be 4.8 ohms when the voltage across it is 8.9 V. This is not correct but still I will use 2.4 ohms load at 8.9 V. This means that the load current on the battery from this circuit is 8.9/2.4 = 3.7 A.
The difference in voltage between the source and the lamp load is 12.7 - 8.9 = 3.8 V. Thus the series resistance from the source to the top of the lamps is 3.8/3.7 = 1.03 ohms. The voltage drop across the one junction connector being measure is small compared to the 3.8 V drop. This junction connector is not the cause of the problem. So you might probe voltages at various points along the top path relative to the top end of the lamps.
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