Distilled water may be slightly more corrosive, since it may contain more Cl- than DI. Per ASTM D1193, DI can produce 18 Meg-ohm water (Type I Reagent Water) with max 1 ppb Cl-, while (single) distillation (Type II) can produce 1 Meg-ohm water with max Cl-.
All high purity water should be protected from air to avoid CO2 absorption.
316L > Al >> Cu for piping. The 316L & Al form more stable passive surface oxide films, while Cu dissolves excessively.