Having read through all the pages of this thread (wish I had found it before I did the repair on my third brake light!) there are a few points that may be useful to future readers.
It may seem obvious to those whose light has simply failed, but the assembly is a complete unit comprising red lens, white or grey rectangular 'container' and its enclosed circuit board. I think at least one person had the lens detach from the container and was thus confused as to how the unit was to be fitted, as you end up with a lens and a container with a metal bracket attached. The temptation is to take the new unit, attach the bracket, then try to work out how on earth this gets fitted! So, whether your unit broke into two parts or is still one, you need to remove the metal bracket in order to fit the new one - unit goes in from the outside, bracket goes on from the inside.
There have been questions raised about whether the lamp is 'tested' on ignition switch-on. Someone commented that the LEDs take so little current that it probably wasn't. True, the LEDs don't take much, but there are internal resistors across the rails to bump the current drawn up to 250mA, which is then tested by the system on switch-on. Without those resistors, the current drawn is only about 80mA which is probably too little. Wasted power, but only when you have your foot on the brake pedal!
Finally, the problem often starts with a crack in the lens. This allows moisture to enter the unit and that can kill some of the LEDs. I replaced 5 LEDs in my unit because they had gone high-resistance, knocking out half the total. I'm going to need a new unit sooner or later, but at least this gets me back on the road!