FYI, a whole family of ions exists: chloride Cl-, hypochlorite ClO-, chlorite ClO2-, chlorate ClO3-, and perchlorate ClO4-.
Electrolysis generates some of the higher ions (mainly chlorate), but how much depends on the pH, concentration, and a few other tweaks. (Chlorate cells normally run at high concentration (starting with saturated brine) and about pH = 6. They emit some chlorine gas, but it's a very minor component -- more than enough to smell, but hardly enough to be worth capturing.)
At high pH, hypochlorite is favored; at low pH, chlorine gas is favored. Hypochlorite may also be favored at low concentration, I don't know?
Probably, pool electrolysis still generates some chlorate, which can be broken down by lowering the pH (which creates the particularly nasty ClO2 gas, but at these concentrations it'll stay fully dissolved; at low pH, it reacts with chloride to make chlorine again). I... don't know if this is actually a recommended procedure for pool maintenance; if it is, then that's the reason.

There are other chemicals that supply "chlorine" (as hypochlorite):
- Plain old bleach is around 6% sodium hypochlorite, with a good 6-8% chloride, the pair being the result of dissolving chlorine gas in NaOH.
- Solid pool chlorine (Pool Shock?) is usually calcium hypochlorite, which, I forget if it's more-or-less pure, or also a mix with CaCl2 (but probably not, because it's stable in air, whereas CaCl2 is hygroscopic), but it's got quite a lot of chlorine content (~30%??).
- The other common option is TCCA (trichlorocyanuric acid), or something similar to it; this is an organic compound, something like chlorine stuck together in solid form with the help of N2 and CO2, which it all returns to when dissolved.
Tim