DI=deionized. RO/DI is so pure that it won't even conduct electricity.
Treating tap water does nothing more than detoxify chlorine, chloramine, heavy metals, etc. It does not remove nitrate, phosphate, silicates, etc. I don't know how well it takes care of heavy metals either, but stuff like copper is the last thing you want in a reef. Although tap water is safe to drink it is a soup of nutrients in a reef system. In nature the water on a reef has very little nutrients, otherwise they turn in to algae farms (which is what we are doing to many of them).
I have never had a single client or customer who has had a nice reef tank using tap water. I have had them try, myself included, but they all come to the same conclusion, it just doesn't work. I am sure there is an exception somewhere out there in the world, but your tank has proven it is not it.
If the distilled water is 100% distilled it should have 0 TDS just like RO/DI. If you can test the TDS/conductivity to verify this you have your solution.