The Zone System as Adams describes it uses spot metering. The goal is to test your materials, so that you can previsualize the relationship between tones in the scene and tones in the print, and get a negative that contains all the information you want to appear in the print. It might require further manipulation at the printing stage, but better to have a negative that's in the ballpark, so you don't have to employ extraordinary measures.
You can spot meter with a handheld meter, or with an in-camera spot metering. Matrix metering tries to do the spot metering automatically, but it makes assumptions about your composition that you may or may not agree with. Spot metering puts you in control.
"Moonrise" required extraordinary measures. Adams based the exposure on the value for the moon--exactly the opposite of the Zone System, which follows the older principle of "expose for the shadows, develop for the highlights." He exposed for the highlights, and then locally intensified the shadows on the neg, and dodged and burned to get everything to look good on the print. On the other hand, he knew that this was probably the best way to get the shot in those days, since the contrast range between the moon and the landscape is huge. Employing the Zone System in the normal way by placing the shadows on Zone III and reducing development for the moon would have produced an even worse negative, because the range would have been too great to compensate for in that way. Today maybe he would have put an ND grad filter into the equation and done it differently.
So for a normal image employing the Zone System, the basic idea is to test the film so that the darkest area where you want detail to show in the film (usually Zone III) always will have detail if you spot meter for Zone III.
Once you've tested for film speed, you test for development time, so that you can spot meter the brightest area where you want to see good detail in the print (usually Zone VIII), and you can adjust the development time so that wherever that area falls, you can make sure it will have detail in the print. If the light is flat, you'll use a longer development time than normal, and if the scene is very contrasty, you'll reduce development time. If you represent the distribution of tones as a histogram, like you might with a digital camera, extending development time stretches the histogram, and reducing development time compresses the histogram.