A garage door cable runs under real tension, wound onto a drum that's connected to a spring doing most of the lifting work. Cutting it loose, or trying to rewind it, without releasing that tension first is how people lose fingers or get hit by a drum spinning loose. This isn't a project where the risk is a bad paint job - it's a project where the risk is a trip to the ER.
If you're set on handling door maintenance yourself, the safe DIY lane is limited: checking for fraying where the cable meets the bottom bracket, listening for a door that runs unevenly (one side higher than the other usually means one cable is stretching or slipping), and keeping the track clear of debris. Beyond that, cable tension work is a job for someone who does it regularly. We won't talk anyone out of a true DIY fix elsewhere, but we also won't pretend a snapped cable on a tensioned drum is a weekend project.