Distilled water can become contaminated when exposed to bacteria so using it in a water bottle is not recommended.
Distilled boiled water cannot be contaminated because there’s no organic material for bacteria to thrive on – so in essence, distilled boiled water is safe for both drinking and storage when the container has airtight seals. However, when this type of sterile solution spills out of its container into your dish, or if it comes in contact with any non-sterile surface at all, that’s where contamination can occur because you are introducing organic matter (i.e., bacteria) back into the sterile distillate (in other words – sorry). Thus while using literally any other type of liquid may contaminate distilled boiled water.