How does the Caipirinha's alcohol ingredient, Cachaca, get classified in here? Is it something completely different? What about Aguardiente?
It's hard to say what the difference is, exactly, other than style of fermentation and distillation.
Despite what importers and makers may want you to believe, cachaça, strictly speaking, is
rum. After all, what is "rum?" Rum is a distilled spirit derived from sugar cane. It is most often made from molasses, a by-product of the sugar refining process, but is also made from sugar cane juice. The rum that is made from sugar cane juice in the French-speaking Caribbean islands is called "rhum agricole" (agricultural rum). The rum that is made from sugar cane juice in Brazil is called "Aguardiente." To my taste, cachaça and rhum agricole have different characteristic qualities that distinguish them, most likely due to different production methods and goals. Then again, Jamacian rum is different from Cuban rum -- but they're still both rum.
Aguardiente is a little more complicated. The word means simply "firey water" and it is apparently made and understood differently depending on where it is made. In Mexico it can include agave, and in Columbia it is flavored with anise, etc. In the case where it is made from sugar cane, and possibly flavored, it would seem appropriate to call it rum or flavored rum. If the wash contains agave or other fermentables, I'm not sure it would be entirely accurate to call it rum.
For sure there is no substituting aguardiente, cachaça and rhum agricole for one another. Strangely, I've found that the closest thing to cachaça in flavor is a rough pisco, which is made from an entirely different fermentable (grapes).