Retired chemist speaking here... wants to clear up some misconceptions about water.
The difference between distilled and reverse osmosis (RO) water is functionally ZERO. Both processes remove both organic and inorganic contaminants from water. They mechanism by which they work is different, the result is the same.
I would be very suprised to find that bottled distilled water is actually made by distillation... the energy costs are very high compared to the RO process. Thus, I would not be surprised at all to find that commercial labeling regulations allow water purified via RO to be labeled as distilled as they are functionally the same.
Deionized water, is likely different than distilled or RO, in that the deionization process does not deal with organic contaminants (nor with residual chlorine in some tap water) it deals only with the charged inorganic components. Or maybe to but it another way distilled water and RO water are deionized plus.
As far as Brita filters go, from looking at their website, these are meant to remove chlorine, many trace organic compounds and some metallic ions, but not sodium from water. The key to using these is to follow the instructions carefully and to keep track of how much water you put through a filter cartridge. The filters have a limited capacity for holding contaminants and there is no easy way to see that a filter has exceeded its capacity.
Regarding the use of any sort of purified water vs. tap water for photographic processes, if you are using have municipal water coming from your taps, it is unlikely that you will see a benefit to using purified water. The contaminants most likely to cause trouble in photo processing are redox active metals, most commonly iron and manganese. Since these elements also cause staining of plumbing fixtures, they are generally well controlled in municipal water systems. If you have a private well, you are probably also OK, unless you know that your water is high in iron or manganese.
My best advise, before routinely going to the expense of using purified water, is to test out your tap water and see if it works in your darkroom. If it works, but you want to be extra cautious, you could use purified water for stock solutions that will be stored for longish intervals and tap water for everything else. Problems with small amounts of contaminants are more likely to show up upon storage.
Lastly, a word about hard water (high calcium and/or magnesium)... this should not be a problem with photographic chemistry as neither metal is redox active. However, hard water, including hard water softened by adding salt, can cause problems with spotting on negatives. The water evaporates leaving the salts behind. In this case, a final rinse with purified water is certainly warranted.
I hope that this is helpful,
--- Frank