I understand that you can either punch or fold dough before shaping it and both have different benefits.
Just understand what you're doing. During the fermentation process (rising), yeast consumes sugar and starch in the dough and turns it into alcohol, carbon dioxide, etc. That carbon dioxide collects gets trapped in the dough (thanks largely to the gluten), and bubbles form. Those bubbles (along with good flavor) are a big reason we ferment dough in the first place, so you want to keep most of them, but you also don't want large bubbles that'll create big empty lumps in your pizza.
"Punching down" the dough and folding it accomplish the same things: the idea is to release the gas from the large bubbles, and also to shift the yeast cells around a bit after they've used up all the food that's available in their immediate vicinities. You don't really have to punch the dough with force like you're angry at it, although some people (especially on TV) like to make a show of doing so. You can just press your hand down into the risen dough so that some of the gas gets pressed out. Removing the dough from its bowl and folding/kneading it a little bit does the same thing, and probably also helps develop the gluten a bit.
This recipe doesn’t mention this step, though. Is it just supposed to be understood that I’ll have to punch it before shaping or do they intend that I somehow divide and shape it without knocking the dough down.
Yes, dividing the dough and forming balls will do exactly the same thing as punching it down, perhaps a bit more gently. Forming balls will also give you a nice round shape that'll turn into a disc as the dough relaxes, and that disc will be easy to form into a nicely shaped pizza.
Given the long fermentation time, I'll bet the recipe you're using calls for a relatively small amount of yeast. Between that and retarding the fermentation in the refrigerator, it may be that you won't get the kind of large bubbles of CO2 that you'd get with more yeast and shorter rise time, and without large bubbles there's no need to punch down the dough.
Also, would I punch the dough down twice? Once when dividing and again right before shaping?
No, there's no need for that. Shaping the pizza will break any large bubbles. Sometimes large bubbles will start to form as you're shaping the pizza, or even after you've put it in the oven, and it's fine to pop those with the tip of a knife if you want to.