With One-Step / Oxiclean. You should definitely rinse afterwards. Not to mention, if you use Oxiclean, make sure you get "Oxiclean Free". It lacks the perfumes and dyes. Both of which can taint the flavors of your wine.
As for don't rinse after "cleaning" because water can ruin it. That doesn't make sense to me you should always sanitize after washing as a standard practice. (rinse, then sanitize. All good!) It's far more important to keep chemicals out of the wine that shouldn't be there. Oxiclean and One-Step are made up of Sodium Percarbonate (baking soda) and Sodium Carbonate. (washing soda) They form oxygen, hydrogen peroxide and soda ash when mixed with water. I don't want (and you shouldn't either!) hydrogen peroxide in my wine.
As for water, I use standard bottled water or bottled spring water. At 88 cents a gallon, it's an easy call.