Watch out with the water softeners. Many of them produce water which is bad for plants, too much sodium. Just like its bad for people
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Reverse Osmosis systems are good since they just take stuff out and don't add anything. The end result from RO can be close to the purity of distilled water but usually not so pure. Most people don't like the taste, or lack of it, of distilled water.
Hard water plus vinegar is a far cry from distilled water. Distilled water contains nothing, maybe a little dissolved carbon dioxide, which is slightly acidic. Hard water plus vinegar contains the end products of the reactions between hard water salts and vinegar. D'uhhh, you knew that right! Those end products are mostly various acetate salts of calcium and magnesium, but some of the original hard water minerals like iron salts may also still be there. You can make your own distilled water for free with a solar still, basically a sheet of glass sloped so that the condensation runs down and is collected.
Rainwater is fairly pure until it hits ground, but will certainly contain as much carbon dioxide as the water can hold plus low levels of other dissolved gases like nitrogen oxides and maybe some sulphates. Some of these are fertilisers, most are slightly acidifying. Rainwater has a pH of about 5.5, lower if you live downwind of Ohio
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Fresh distilled water has a pH of 7 but it fairly quickly drops to around 5.5 if left open to the air.