Are the two strings connected together at each end, or do you have a dual-output driver?
If they were connected in parallel and you shorted one LED, I would expect that the other two in that string should light with twice the current, and the other string to go out, or nearly so.
This is because the driver should have a fixed current output, not a fixed voltage. Let's say your driver is set to 1A, and you expect 0.5A in each string. The LEDs will have a voltage vs. current curve, which might go through 2.2V at 5 mA, 3.0V at 500 mA, and 3.3V at 1000 mA. This is actually pretty close to realistic.
So if you have two strings of three, the total voltage would be 9V. If you had only two strings of two, the voltage would be 6V. But if you had only one string of 2, it would be 6.6V, because the LED voltage would be 3.3V at 1A. Now add a string of three in parallel with the string of 2. 6.6/2 = 2.2V, so the LEDs in the string of 3 would draw only 5 mA. They'd be so dim you wouldn't see them next to the other string. This is exactly what you have when you short out one of the LEDs.
Of course, if the driver has 2 separate outputs, then everything changes, as it's really like 2 separate drivers running 2 separate strings of LEDs. In that case, it's possible the driver doesn't like the string of 2, and shuts itself off to protect itself from overheating.