There are a lot of opinions on what kind of water to drink so I will leave answering that question to those people who are adamant about one type or another.

But, I can go about answering the question about what distilled water is. It's pretty basic. Distilled water is water that has been brought to a gaseous state (through boiling or other means) and then condensed. The condensation shouldn't be mixed with the original source, otherwise it will not be "distilled."

I am thinking of a suvival story I watched on tv. A guy got stranded in the atlantic ocean on a life raft after his boat sunk. He was on that raft for weeks, if not months. Of course the water available would not have been so great to quench his thirst as it was loaded with salt. He had a little contraption that was something like a double boiler, the top pot was smaller than the bottom one. To get distilled water, he would place water in the top pot, put that in the larger pot, which had some means to float. The whole thing was placed in plastic and set to float by the raft during the day. The sunlight would heat the water and the gaseous water would condense on the plastic. After condensing, the water would run down the sides of the plastic into the larger pot and would be free of salt, thus giving him a source of potable water.

Distillation is a refining process, as used in oil refineries to produce petroleum products.
Therefore distilled water is REFINED water and contains no minerals.