I was a cable builder for 13 years.
dd's explanation/diagram was spot on.
The only tip's I might add is that they make a waterproof shrink tubing that has a glue/sealer inside. I/we would also use a short section of medical grade silicone tubing that we would put over the solder connection and then use the water proof shrink tubing cut a 1/2" longer than silicon tubing. The silicone adds a layer of protection at the solder joint and replaces the wire coating you lose when you strip back the ends to splice/solder. In event of any friction in wire lengths in finished cable, the silicone will prevent any abrasion at the solder joint.
We did this for cables for military apps.
Just a upgrade, not an necessity.
Putting the shielding back is a necessity. When you finish your splice, the cable should be as original as when you started for maximum efficiency.