Just a guess but, I think Elon is planning on distilling the moisture out of filtered air which is passed over the heat pump condenser coils.Each to his own, of course but distilled water is not pure. That's why if you want pure water in your lab you don't use distilled and if you want pure water from the pharmacy you buy triple distilled. If you want pure water today you go with RO. The cartridges seem to last forever (depending on volume processed - I'm on the original set I got over 15 yrs ago and they are still producing 2 -3 ppm water) but of course only reject about 98% of most minerals and the particulate and carbon filters, there to protect the membranes, are also great for removing bacteria, odors etc. In the CT application there would be additional filters for removing viruses, cryptosporidium cysts etc. If really pure water is wanted RO (which gets typical tap water down to a couple ppm) is followed by ion exchange and that's what labs, reef aquarists, dialysis centers, semiconductor fabricators, brewers etc. do these days.
The main reason that the CT, if it contains water purification gear at all, will not use distillation is energy. It takes 2257 kJ (627 Wh) of energy to boil a litre of water. To get to equivalent to RO purity you'd have to do that 3 times for a total of about 1.9 kWh. To prepare 1 L of RO water my system (I wash my cars with RO) uses 24 kJ (6.67 Wh). It has a 1/2 HP pump.
I hope this is one of those Elon ideas that actually gets implemented! If I could be out overlanding and have a gallon or 2 of water at the end of the day, that would be a real game changer. Right now, the amount of water I can carry can be the most limiting factor on a trip.