I am not a chemist but I have read that distilled water may actually be lacking in some of the trace elements that are good for wine making. I use bottled water that I buy in gallon jugs. I would use my deep well water but it actually has a lot of iron in it.
I haven't made kits, just from grapes or fresh juice so I don't use water expect for dissolving additions . When I do that, I use reverse osmosis water if I'm doing something larger like making syrup for back sweetening. If I am just dissolving some sulfites I use tap water, at that concentration it will kill anything anyway.
I would never use distilled water as other mentioned - some trace minerals and such aid the process.
As to bottled water - there is always the 'chance' that the plastic could leach out something into your water. I have been using screw-top plastic jugs that I got water in from Aldi's BUT I noticed when I refill them at Walmart machine ($ 0.39 / gallon) that they use city sourced tap water filtered and hit with UV light. Still some chlorine and fluoride in it then I guess.
At some point you just have to decide what you are comfortable with. I suppose if I was concerned about leaching chemicals from plastic jugs I could take in my extra glass carboys and fill those but then there is still the fluoride and chlorine issue.
While Farcical I always think of a "Doc Martin" episode (Season 1 Episode 3) when I hear the words "Bottled Spring Water"
(The entrepreneurial character "Bert Large" attempts to make money by bottling 'spring' water and hawking it to local townsfolks. Not by coincidence there is an outbreak of intestinal 'disturbances' which eventually traces back to the bottled spring water.)
Not to cast doubt on spring water sources but... nowdays there seems to be one surprise after another as entrepreneurs are revealed to be less than upfront and honest about their businesses.