Rum has a very distinct taste from cachaca,

For the reason already explained, saying "rum has a very distinct taste from cachaça" is as silly as saying that "fruits have a very distinct taste from apples", or that "fermented Ives Noir has a very distinct taste from wine". One includes the other; all tastes of one will be included as tastes of the other, by definition. Like this:

The only reason why people claim that cachaça is "not rum" is nationalists shilling exoticism, a long time ago. They screeched so much when you called a duck "duck" that people started pretending that the duck is "magically" different if it quacks in Vargas' Reich. That's it. In the meantime, other cultivated rums kept being called "rum". (The Australian rum that I mentioned in the Venn diagram is an example. It tastes... well, like cachaça.)

Now, on cachaça and other types of rum (yup) tasting differently: some will be extremely similar, some will be completely unlike each other. A simple rum will taste almost the same as a non-aged Velho Barreiro, even if the rum in question is made of molasses, like a "simple" Bacardi. And both will taste completely alien compared with an Anísio Santiago (one of those expensive cachaças from Salinas), even if we both agree that the later is cachaça (and likely that it's a poor choice for caipirinha). What affects flavour the most isn't even if it's cachaça or another rum, but it's how it's handled past distillation, and for drinks you'll probably pick the simplest one anyway.