In summary:
Protein skimmer removes large organic molecule with certain properties in saltwater environment.
Tap Water Purifier (TWP) is a bare-bone DI filter that works well if water demand is low, and input water quality is high. Not a good alternative if you are in this hobby for a long term.
DI removes any charged ion. Almost all contaminants in water is charged.
RO removes contaminants base on size. It is not effiective for small molecules similar to H2O.
Water softener replace calcium ion in water with sodium. The net amount of ion is the same after treatment, but sodium is considered "soft," hence the name, water softener.
For ultrapure water, nothing beat RO/DI, not even distillation.