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?
The flat disc is a bit more precise in determining the lighting ratio of the lamps themselves, as it reads a narrower field. It is not influenced by the other lights as much. It more truly reads just the lamps themselves, while the dome will give you a better idea of how the light is actually falling on the face. If there is flare/spill from one lamp to the area mostly lit by another lamp, the dome will read that, while the disc will not. IMO, you want to meter the flare, as it is what you will actually capture when you shoot.
In your example, to capture the lighting ratio just like you designed it so that it prints as intended at your normal printing time and filter, you would expose for the brighter lamp number two (which I would actually call lamp number one, or the main light). If you expose by pointing the lamp at the main light, then the fill light will be captured at the intended value, not too brightly, like with the averaging/toward-the-camera method.
Now, if you were stuck with a certain lighting ratio (natural light, e.g.), and you would prefer if it were less contrasty on the print than it is in reality, that is the time to either average down the middle or meter the darker side, and probably adjust development as well.
In practice if you don't have a flat plane receptor you can shield the incidental light dome with your hand, or make a shallow cardboard tube that fits over it so it only takes frontal light into account..
The lighting on the main, the most important side of the subject, needs to match the camera setting.
That camera setting places the important parts of the subject's "zones and tones" in all the intended places.
At that point your camera and main light settings, are "etched in stone".
The measurement and adjustment of lighting on the the other side of the subject is done "after-the-fact" to control the ratio.
If the light source illuminating the shadows on one side of the subject is in the nature of a diffused "fill" light, it will also be contributing illumination to the other, highlight side. Thus a change to that fill light will not only change the ratio, it will also necessitate a change to the camera exposure settings.
So it is important to differentiate between lighting setups that have two sources that illuminate separate sides, from lighting setups where one source is more of an all-over "fill".
Maybe an example would help.
(Just FYI I use strobes.)
First, I decide on the DOF I want and set the aperture that will get me what I want, say f4 a classic short DOF portrait. This does not change once it has been set.
Second, I measure ambient light and set the shutter speed to get exactly the amount of ambient light I want, say 1/60th to allow the back ground in or 1/400th to keep the background out. (With strobes, as long as we are in the sync range, shutter speed has no effect on the main subject's exposure.) This setting also does not change once set.
Third, I measure and adjust the main strobe to match the camera aperture setting (f4). This is then left alone.
Forth, I measure and adjust the secondary strobe to get me the ratio I want, maybe f4 for a 1:1 fashion shot, f2.8 for a gentle 2:1, or f2 for 4:1.
Turn on main light, point the dome at it and take a reading. Turn off main light.
Turn on fill light, point the dome at it and take a reading.
For a 2:1 ratio, fill should meter one stop less light than main.
Turn main light back on. point the dome at the camera, and take a reading.
Set the camera and take a picture.
More complex lighting setups will call for more complex metering procedures, hence the need for the flat panel.