Oops, time for some clarification. I was certainly unclear, and maybe going the wrong direction on my mental meter it seems...
So, to summarize:
If I spot meter, I read a shadow and then place it in Zone III by stopping down two stops from the indicated reading. If I use an averaging meter to read the same shadow, e.g., by getting very close to the subject I'm metering so that other luminance values won't affect the reading, then I do the same. My spot meters, however, have Zone dials/stickers on them, so I no longer think of metering and placing my Zone III (or VI) value as "stopping down."
However, if I use an averaging reflected light meter and meter the entire scene, and then assume this average is middle gray, I use the indicated reading and hope that the average really is middle gray (whatever that means to my meter, Bill and Steve

) and that the shadows I want detail in are really two stops down from that average and fall nicely in Zone III. If I use a gray card, I do the same. This method works well in all but contrasty situations, but can give more exposure than needed in flat light, especially if expansions are being applied.
If I take an incident reading in the shadows and use the indicated settings, I'll get good shadow detail, but may have more exposure than really needed. No problem for most negative materials, but this could lead to overexposure and subsequent highlight blocking in really contrasty situations and/or more grain with small film.
If I take an incident reading in the light that is illuminating the non-shadowed areas of my scene, I'll have a great reading for slide film (as mentioned above), but will lose shadows in contrasty situations, which is not optimum for negative materials. Often this is where advice is given to open up some from the meter reading, either by an arbitrary stop or so, or by taking a shadow reading and averaging. Both of these latter likely lead to less-than-ideal exposure in more extreme situations.
What Minor White seems to be doing in your example Bill is "placing" his shadowed palm in Zone IV, which would put his sunlit palm in Zone VI
if there was a two-stop difference in the illumination between lighted and shaded areas of the scene. This is pretty common and, as mentioned above, one often hears "take a reading in the light and open up two stops for subjects with lit and shaded areas when one just takes an incident reading in the light. This could be the source of my open-two-stops from the meter reading unclearness. Again, I think this works for many averagely-lit scenes, but not for extreme situations.
There, I think I've got it in a clearer form now. Sorry for the confusion.
Best,
Doremus