I have it on good authority that the Sun (capitalized) is the correct English proper noun for the specific primary of the solar system we inhabit. “sun” refers to any star, or to various other meanings (“got too much sun”). “sun” can also be used when speaking casually about the specific star, but “Sun” is used to distinguish it from all other stars. “Sol” is mostly used in science-fiction.
Similarly, “Earth” is the planet to which science fiction writers refer as Sol III. “earth” is soil, and “the earth” is the planet in most contexts. Same goes for “Moon”, “Universe”, “Solar system”, and “Asteroid Belt”.