Yea, we are basically consuming distilled water, most of the time.
I guess that really depends on where in the world you live though. Here in Norway most of the water comes straight from groundwater resources with no or very little processing before we drink it from the tap. I haven't seen it analyzed, but I would assume that this has a pretty nice mineral concentration compared to battled water.
Anyway, interesting topic you bring up @exercisinghealth :)