A lot of mixed up not-quite-chemistry here. Just to keep things in perspective, the salt (NaCl) concentration in a SWG pool is around 3300 ppm which is in the form of chloride ions, a negatively charged form the chlorine atom. You're going to have to add a lot of bleach to reach that level. For each ppm of chlorine you add via bleach (NaOCl), you will eventually get 1 ppm of chloride (NaCl), more or less. Keep in mind that every time it rains, it will reduce the concentration of NaCl that has been building up since the water will become diluted.
I tried to find out if salt were intentionally added to bleach for some reason and it doesn't appear to be. It is probably there because bleach degrades over time and the primary mechanism is decomposition into oxygen and salt:
2NaOCl --> O2 + 2NaCl
Note that if you keep your pool acidic, you may have degradation to chlorine gas instead.
The way that bleach works is that the form of chlorine in bleach (non-charged) is incredibly reactive and rip apart almost anything it interacts with by stripping off a single electron. The end result is that the chlorine (uncharged) is converted to chloride (negative charge).
For me, with a SWG pool, I completely ignore the chloride added when I use bleach. (I actually use it quite often since it is more efficient to add chlorine after an afternoon of kids via bleach than cranking up the SWG.)
Specifically, in answer to this:
The correct way to explain it is that all forms of chlorine typically put into pools contain salt. The chlorine part gets used up, but the salt remains. Correct?
No, although there may be salt in the bleach (see above) which can contribute to the chloride level in the pool, what happens is that the chlorine in the bleach does its job, and in return gets converted to chloride (the form of chlorine in salt.)
If you have a SWG, the chloride (charged) gets recycled back to chlorine (uncharged) with the help of electricity. This is how bleach is actually made.
This is pretty basic chemistry. Let me guess, the son was pre-med.