Moss, I can 鍍est your theory of the second diagram. As I mentioned earlier, when I unplug the power from the wall receptacle and test the float switch leads, there is no continuity and of course, this is what we want, and tells us that the float in the pump tank buried outside under the snow is NOT in an alarm condition.
But if your second diagram is right, I should get continuity at the float switch leads when I turn the test switch on.
Of course, that would indicate that everything os A OK on the box, which we know is not the case. (The alarm goes on as soon as I plug un the power.)
So it may be the test switch with some sort of weird intermittent problem, but I think it is the silence switch if I had to bet money on it.
A fun puzzle to work on, because fortunately the pump in the pump tank is functioning properly during all this time!