Nice explanation and almost correct. High quality deionized water is almost completely unbuffered and as you say will absorb anything it comes in contact with such as gases in the atmosphere or metals from a container. Most distilled water is made by, well, distillation. Distillation is less efficient since small droplets get entrained in the flow of steam and some minerals are carried over. But still pretty much the same as DI for this discussion.

Reverse-osmosis or RO is different. The water is forced thru a membrane by pressure. The membrane passes a portion of the minerals thru so it is not as pure as deionized or distilled water. In home systems RO typically reduces TDS (total dissolved solids) by about 85% or so. RO water falls into the high quality tap water category so use it if you have it.

On the practical side, assuming you have reasonable quality tap water, you can use it for both diluting coolant and adding to a battery (MB's recommendation). You can use DI water in the coolant because antifreeze is more than just ethylene glycol. It contains corrosion inhibitors which provide the buffering. Using DI water in a battery might make it last a little longer if you measure the battery life with a stopwatch. :D But if you are adding water regularly, it's time to check out your charging system.

As for the windshield/headlamp washing system, a high purity water will leave less streaks and less clogging of the lines/nozzles.

In theory.

It probably though has picked up 'stuff' from the surfaces it has touched.

If you want pure water I'd spend $2 and buy a bottle.

Instead of chemistry lessons, which may or may to apply since we aren't discussing using 100% distilled water in the cooling system in lieu of mixing it with coolant:

Mercedes-Benz Service Microfiche System, 4/93:

Use water which is clean and not too hard. Drinking water frequently, but not always, satisfies requirements. The content of dissolved substances in the water may be of significance for the occurrence of corrosion. If the water quality is not known, DISTILLED or fully desalinated water should be
used. For water quality see Mercedes-Benz Specifications for Service Products.

Mercedes-Benz Specifications for Service Products, Section 310:

Fresh water quality with a coolant composition according to item (engines for passenger cars and commercial vehicles)

Sum of alkaline earth (water hardness): 0 to 3.6 mmol/l (0 to 20 degrees d) pH value at 20 degrees C: 6.5 to 8.5

Selling its cars worldwide requires some specificity, since water quality varies so greatly around the planet. Follow the recommendations and renew the coolant every other year and your cooling system will be fine.