The concept is that the ultra-pure water leaches potentially harmful ions from its surroundings, for example chromium from stainless steel fixtures. Better to drink some harmless sodium and chlorine ions.
Distilled water is boiled and recondensed water, which leaves you with pure H2O (no minerals etc). It's safe to drink but pretty bland.
And DI water gets hydronium and hydroxide pretty quickly (if they ever go away.) H2O just breaks down into OH- and H3O+ on its own (as far as I know)
>Deionization is a chemical process that uses specially manufactured ion-exchange resins, which exchange hydrogen and hydroxide ions for dissolved minerals
https://en.wikipedia.org/wiki/Purified_water#Deionization
In other words, if deionized water is unhealthful because of a lack of ions, distilled water is, too, because it doesn't contain any more ions than deionized water does.
A friend bought a water destiller because of some health-theories. You are supposed to drink distilled water in the morning and so on... He is usually using it on tap water because he believes the water isn't pure enough (Germany, should be pure enough). One day he figured he could try the bottled water, maybe less build up of chalk in the sistern. This area has a lot of chalk in the ground so everything gets white very fast. He was not totally surpriced with having a black burned, smelly gunk in the bottom the next day. He does not buy bottled water any more :-)