Rain water, which is what arboreal chameleons have evolved on, begins life as evaporate from oceans, freshwater, and transpiration from plants. It does not contain any minerals at all. As in zero. It begins at neutral pH 7.0, but very quickly, atmospheric CO2 dissolves into it and a very, very weak carbonic acid solution brings the pH ever so slightly into "acidic" (~5.5). In our modern day atmosphere water may pick up pollutants depending on where it condensed and its contact with industrial emissions - but we are not talking about using rainwater, only "what is normal natural for chameleons during their evolution and what is closest to this that we can supply."
Deionized water is a term for water which has had nearly every trace of minerals and organic molecules removed. It is, in a properly functioning system, nearly identical to:
Distilled water is water that has been evaporated, then recondensed to its pure form. There is no difference between pure rain water and pure distilled water (although they can become different as they contact pipes, tubes, leaves, branches, dust).
Spring water varies in its profile depending on the rocks and minerals it has come in contact with. It can vary from deadly poisonous to nearly "pure"
Tap water varies so much from county to county and even from house to house, that to say that all tap water is fine to consume (by pets or people) is like saying all shellfish is always fine to eat, regardless of where it was harvested, or how it smells.
Most of what is said above and even by vets and health food fanatics is anecdotal, unscientific, and highly biased by one's personal beliefs. I do not believe blanket statements can be made about water quality from different sources.
I do know that chameleons evolved on rainwater for the most part, and I mimic that with RO water in my system. I also know that, in saltwater aquaria, the organisms are extremely sensitive to ppm quantities of lead, copper, chlorine, and other elements. Chameleons may or may not be sensitive to these, but if a study were conducted, I would place my money on the possibility that they are. With the most compelling argument being that, at the very least, chlorine would be irritating to sensitive mucous membranes of the eyes and sinuses and would possibly disrupt the balance of beneficial flora of those areas, thus increasing the possibility of infections. This effect could also result from highly mineralized, or slightly to very akaline water.
Finally, none of this is to say that high quality tap water (neutral ph, maybe filtered through a simple charcoal filter) would not be perfectly adequate for chameleons. But it is my experience that most tap water is akaline and contains significant levels of chlorine and chloramines. And while the chlorine/chloramines can be removed with charcoal, the alkalinity would remain as an irritant.
One of the greatest joys of keeping these creatures for me, is the process of refining my husbandry and recreating a "natural setting." I have so, so much to learn in this regard. But, like most of you, I am committed to becoming a better keeper and providing the healthiest parameters for my animals. To me, water quality is an important piece of the puzzle.