First some definitions:
Sol is short for Solar day, that is, the time it takes to get from a point on day one to the same point on day two, based on the Earth's motion relative to the sun (i.e., from the point where the sun is directly overhead one day to the sun in the same position the next day). What is usually used is the average over time, as the length of a solar day can vary (and does, on Earth and Mars, due to axial tilt and elliptical orbits). One mean solar day is 24 hours on Eath, and 24 hours, 39 minutes, 35.2 seconds on Mars.
A Sidereal day, on the other hand, is the time it takes to get from a point on day one to the same point on day two, based on the Earth's motion relative to the stars. It is the time it takes a planet to rotate completely around its axis. It likewise varies. One mean sidereal day on Earth is 23 hours, 56 minutes, 4.1 seconds; on Mars is 24 hours, 37 minutes, 22 seconds.
Sol is the Latin name for the sun (and some other languages derived from Latin), but context should quickly distinguish the two meanings; I can think of no examples (that are not needlessly convoluted) where the two could be mixed up in usage.
The shorter 'Sol' could also be used for the solar days of other planets rather than the full 'Solar day', but is not likely to be, just to avoid confusion. (Mars got dibs on that one!) It might be used with the name of the planet or moon to sufficiently distinguish it. So far the issue hasn't come up, as there is rarely need to reference the length of days on other planets. It only arose with Mars because of the rovers and probes that have been sent there that operate on solar power, making it more convenient for the people who worked with them to operate on the same solar schedule that Mars does. Humans being humans, "Solar day" got shortened to "Sol", and after enough use that became the recognized word for a solar day on Mars (and 'yestersol' as the solar day prior to the current Sol).