Distilled water id pure water, nothing but H2O. All other waters contain some particulates, usually inorganic, some dangerous.
Avoid tap water all you can, it contains chemicals, inorganic minerals. and occasionally parasites and bacteria.
Filtered water goes through an activated charcoal filter and has had most particulates and odors removed. It does not remove all chemicals, for example, filtering will remove chlorine taste, but not the chemical itself.
Well and spring water can be good or bad. Most of this water is hard water containing a lot of inorganic minerals. Well and spring water can also absorb toxic chemicals that have been dumped on the ground including carcinogens. Both waters can also contain bacteria and parasites, and, in some cases, animal waste. If you use well or local spring water have it checked out by a reputable lab.
Reverse Osmosis water has been filtered through a series of ultra fine screens. It is pure for the most part. Some claim RO water changes the structure of water making it unfit to drink. but this is a theory. RO water is usually readily available at supermarkets and even health food stores. It can be purchased pre-bottled and as refills.
Bottled distilled water can leach out plastic from its containers. There is also some question about the purity and cleanliness of its bottling process.
Home steam distilled water is very pure. Water stored in plastic should be used quickly, but when distilled and stored in glass jugs it can last a long time. Despite some claims, distilled water will not leach good minerals from the body, however it can leach out inorganic minerals stored in joints and other tissue. Home distilled water is the very best water to use while cleansing. Many of us drink it all the time.