A carbon filter would get rid of the organics and low-boiling point things.
So optimal water purification would first use a carbon filter, and then distill the output of that. Make sure to add minerals to the final product though. Pure water is a vicious solvent and will slowly melt your teeth away. It will also become quite acidic after sucking CO2 out of the air, so consider adding some pH balancing salts too.
I suspect that this one is identical for all practical purposes.
Got a source for "really bad"? Because that would conflict with the above link. I stand by my comment above: distilled water, passed through a carbon-filter post-distilation, seems like a better bet than drinking water the city pumps out of the ground in Sili Valley.
Those are not the only two options. There is a third option that exists, it's called purified water and it leaves the important minerals and salts in the water. Water can be purified without being distilled, and even your own source says distilled water will leach chemicals from the plastic container into the water. Distilled water is what you put into a car's radiator or a home humidifier, not into your body.
There are plenty of sources from actual medical journals and the WHO (not random Internet blogs) on the negative health impacts of distilled water here: https://en.wikipedia.org/wiki/Distilled_water#Health_effects