This has been hashed and re-hashed many times on many forums over the years and some will argue with the following, but... well take it for what it is... my understanding of the subject...
Distilled water is water that has been changed to a gas (steam) by heating and then condensed to water again in a different container... NOTE: just boiling tap water does NOT make it distilled... all that does is concentrate the dissolved minerals by removing some of the H2O in steam that escapes... (it might make the water "sterile" and your boiler will not get the flue or salmonella poisoning, but it leaves the carcasses of the beasties in the water to form limestone on the insides of the boiler). You have to capture the steam that has left the pot and condense it back to water to get "Distilled water". You also need to NOT capture the first of the "steam" as it might contain other chemicals that vaporize before the H2O does.
"Deionized water" is NOT "Distilled water". Water is normally slightly ionized -- electrically charged due to the water molecules either missing an electron or having an extra one (thus the term "ion" and "deionized" means removing the ions). When you neutralize the charge, the nature of the water will be to pull ions from surrounding material, Copper, brass (copper and zinc) and bronze (copper and tin) being metals, freely give up atoms in deionized water... thus your boiler and fittings will become thinner and porous (especially the brass and bronze fittings as zinc and tin will leach out of the molecular bonds and dissolve in the deionized water. Bronze fittings have been known to be easily crushed after a few years in deionized water due to the loss of the tin... grip an old bronze fitting with a wrench or pliers and it will break into many small dusty pieces!
Rain water is a pretty good source of distilled water. But it depends on how you capture it. If it has run off your roof, where the birdies and other beasties of the air have left "deposits",and leaves and other dust and debris are, then there will be germs and other contaminates in the water that should be filtered out. Not that the "germs" are a problem, but their boiled dead bodies will help form the insulating film on the inside of the boiler that you don't want to form.
Water from your dehumidifier is a pretty good source of distilled water. It will be mostly H2O, but since it is collected from free air it will also contain mold spores, pollen, dust, cooking oils and anything else that can get airborne and then collect on the cooling fins of the dehumidifier and get washed into the holding tank. A fine filter could remove the larger stuff, and oils and such could be skimmed off, but that may not be necessary as the amount of these materials would normally be quite low.
Relatively pure H2O is also made with "Reverse Osmosis", where the water passes through a membrane... this is similar to just using an extremely fine filter, but it isn't just "filtered water". The membrane passes only the H2O on a molecular basis, so minerals (and larger contaminants) don't normally make it through. Some oils and other liquids can get through so it is not the same as Distilled water, but it is very close.
"Soft water" is NOT a substitute for Distilled water. It is only "Hard water" that has the dissolved calcium replaced with sodium. It may be only slightly better than plain tap water but will still leave minerals in the boiler to form the insulating film.
If you would like to see what is being left in your boiler, fill a pot with whatever water you want to use and put it on the stove and boil it all away... WATCH THE POT so it does not melt on the stove when the water is gone!!!! Do several gallons this way without rinsing the pot. If a whitish film is left in the pot, you have water with minerals in it and it is not a good water to use in your EXPENSIVE toy boiler.
EDIT: daing spellchecker changed "zink" to "ink", when I thought it was changing to "zinc"