People have different approaches for arriving at the same result. Barnbaum places his shadows high (exposing more fully), "develops less" (than he was when trying to get shadow detail out of an underexposed negative) and gets what he wants. Some simply rate the film slower (exposing more fully, i.e., getting Zone III up on the straight line portion of their film) and test for an optimum development time (likely less than manufacturers' recommendations) and get basically the same result.
Getting shadow detail on the straight-line portion of the curve is a good idea for a lot of photography. For other styles, it's less important (think portraiture, where the mid-tones are most important, or the "pushed" look, which dumps shadows entirely in favor of a contrasty rendering). Some films really need a lot of "extra" exposure to get Zone III up on the straight-line portion of the film. 320Tri-X is such a film. Others, like the T-Max films, have a straight-line portion that goes down very low and a very short toe section. Deciding what exposure to give, then, depends on 1) where you want your shadows in the first place, i.e., on the straight-line portion or not, and 2) the characteristics of the film you are using.
Barnbaum is generally correct, however, in that an underexposed negative with less separation in the shadows than one wants (whatever that is) gives lackluster prints. My advice to students who are having trouble getting satisfactory prints is almost always, "expose your film more."
The basic exposure and development part of the Zone System aims to get printable detail in the darkest shadows (i.e., adequate exposure) and put the highlights in a place where they are not difficult to print (proper development). At it's most basic level, then, it's simply: expose for the shadows and develop for the highlights. How you usually find the right exposure for the shadows is to do a test at different exposures, make prints and choose the exposure that gives you the shadow detail and separation you want. You can call this a personal film speed, place your shadows in Zone VI, dial in exposure compensation on your automatic camera, whatever you like. The Zone System likes you to think in terms of a personal film speed (E.I. = exposure index) based on your testing. Like the negative that's 2/3 underexposed from box speed, then rate your film 2/3 stop slower.
Finding the right development is second. Does the highlight fall too low? Expansion development is needed and vice-versa. If the highlight falls just where we want it, then Normal development is called for. To test, make some identical negatives at your new exposure and develop them for different times. Find the one that prints both the shadows and the highlights where you want them at a contrast setting right in the middle of your contrast range. That's the development for scenes with that brightness range. Do this for a few scenes with different brightness ranges and you've got a choice of development times for different situations.
You don't really need the Zone System to do all this, however. The real advantage of the Zone System is as a tool for visualizing the final print and making your exposure and development choices based on that rather than just making a standard, plain vanilla, "printable" negative. For many, however, this is all the Zone System they need. Learning how to meter effectively and getting all the information they want on the negative lets them print it "any way they want."
Expanding on the visualization aspect, however, many of us decide more precisely where to place a certain shadow value depending on how we want it to look in the print. A dark shadow that still retains texture? That's the classic Zone III placement. Bright shadows in a snow scene? That's Zone V for me. Inky black, solid-looking shadows on a dune? I'll place them on Zone II or even Zone I. This is now using the Zone System as a visualization tool for achieving the desired results in the final print.
Dealing with highlights is the same. Nice white with a modicum of texture still? Classic Zone VIII, I'll choose a development time to put it there, based on the subject brightness range and where I've placed the shadow. But maybe I want a really bright, almost featureless white, or a richer, more detailed white. I'll adjust accordingly.
Nowadays, one can deal with many of the contrast scenarios with darkroom controls instead of changing development. Many do this, especially roll-film users. They find one development time for everything that lets them deal with all the extremes of negative contrasts, low to high, with contrast settings, etc. I use a combination of the two.
Once one has a basic understanding of how the system works, getting a printable negative is easy: avoid underexposure by exposing for a shadow value and find a development time that lets you get the print you want. After that, one can graduate to the more advanced aspects of the system: visualizing the values and breaking the rules for a "normal" negative in order to get the effect one desires. That's the whole reason the Zone System has zones; so we can picture them in our mind's eye.
Best,
Doremus
Face slap! 
Stupid me! 
I have been doing exactly that for several decades with great results! Now I will have to stop doing that just because Witold said I am wrong! 
Woe is me! 
