Hmm, what a problematic manga/appraisal skill. It suggests that distilled water is better quality than boiled, meaning that it implies it is better to drink distilled water.
This is definitely not the case. While you definitely can drink distilled water mostly safely, the lack of minerals and salts/electrolytes means that there are dangers involved. One is that it is far from as effective in hydrating your body, and over long period of usage could probably (this is just my uneducated guess) drain your body of those things if you don't supplement it from other sources. It has also been linked to nutrient deficiencies (most likely for same reason as what I guessed before. Or rather, aren't those two things actually the same thing?). Also likely to damage specific parts of your body when in contact with them, due to lacking any saturation of contaminants. Meaning it is very prone to leeching/extracting contaminants and can leave parts of your body deprived of ie. required salts (eyes?) or flouride (teeth), which btw is also part of why it is ideal for washing most chemical accidents in lab-settings, like if you spilled something into your eyes. The lack of saturation can also make it dissolve more of pollutants that are worse than found in regular tap water, like say, if you were to bottle it in a toxic container (if it is the kind of container that is known to be bad to store tap water in over a long period of time, then that is probably bad for storing distilled water in over shorter periods too).
... There is also a source that says it might have excess contaminants, such as arsenic, but I don't know if that is true as naively I would assume distillation would get rid of that as well as any other contaminant with a boiling-point below waters.