Why do some recipes call for the pot to be covered, others for it to be partially covered or uncovered, while some recipes don't say anything about covering the pot? You've uncovered a minefield for recipe writers. While some struggle to tell readers exactly what to do with their pot lids, others leave readers to figure it out. Here are some guidelines:
Cover a pot whenever you are attempting to bring the contents to a boil. This means covering the pot when you're heating water for pasta - or bringing the water back to a boil after you've added the pasta, or bringing a soup to an initial simmer. A covered pot boils faster than an uncovered one because the cooling presence of the room's atmosphere is greatly diminished.
Once the liquid comes to a boil, the options widen. With placement of the lid, you are attempting to juggle the competing considerations of boil-over, sufficient heat and evaporation. One scenario that requires a covered pot is cooking grains such as rice. Here your goal is not to evaporate the simmering liquid, but to get the rice to absorb it.