Actually churchhaze, it is more than water. That is the problem.
Distilled water: Boiled water that is condensed and collected. Good at removing inorganic salts, not so good at removing many dissolved organics (b.p. at or below that of water, 100oC). Most commercial producers use carbon filters to remove organics. TDS can be as low as 0ppm.
R/O water: Usually filtered through multiple stages, may include UV or other bactericide treatment. Filtration usually includes carbon filters to remove organics as well as a 0.002micron filter than passes water, but very little else. TDS usually in the range of 15-50ppm, depending on source and contaminants.
Deionized water: Technically, both Distilled and R/O water could be considered "deionized." Ion exchange is the other method for producing it, but the exchange filter must be replaced/recharged frequently to remain effective.
In all of these methods the source of the water is irrelevant, provided that all solids have been removed. Our well water frequently is over 1200ppm and smells. After R/O treatment, the ppm is down to 40 and actually drinkable. The thing is, if you start with good water with low TDS, you know what you are mixing and giving to plants (and yourself). It is a good base from which to start. BTW, all water purification methods practically available produce waste water. That can't be avoided. You either have to pony up, find a free source of water to start with, or just go buy your water from a supermarket.