Brake lights flicker then go out. Bad switch? Bad Relay?

[socratic]

When I step lightly on the brake pedal, the brake lights come on but when I step on them a bit harder (less than even the normal pressure to stop the car), the lights go out.

It may be a faulty switch, I took it out, took it apart and cleaned it (there was some grease in there it seems). But the situation remains.

In another thread, a helpful poster suggested I swap in the clutch switch as a test. I did. When I have the clutch switch swapped in, the brake lights stay on all the time, even when the brake pedal is not pressed. Perhaps the clutch switch is slightly different in operation. At least this suggests that the wiring to the brake lights is sound.

I'm going to replace the brake switch anyways since I've monkeyed with it and no longer trust it.

But could there be anything else? Is there such a thing as a brake light RELAY that could be going bad? If so, where is the relay located?

While I'm asking, what is the clutch switch for anyways? I see that I can start the car even though the switch is not there, so it's not the clutch-ignition cut-out.