Let's say I have two lamps. By using the dome, and pointing from both sides of the face to the camera, I get those values:

I lamp: 1/125, f/8;
II lamp: 1/125, f/11;

Those measures tell me both what the average exposure would be (1/125, f/9.5) and the illuminance ratio (2:1).
So why would somebody need the flat thing if the with the dome thing one can kill both birds with one stone?

The dome can be used for ratio inspection. Shouldn't there be some application in which the flat thing is necessary? Or more reliable? Which is the reason for the flat thing to exist?