Distilled water is neutral, but natural surface water with low mineral content still has CO2 disolved in it, and usually also some SO2 and NOx, which makes the water acidic, so what's wrong is the comparison with distilled water, not that it is acidic.
I should have expressed this more carefully. Distilled water is by itself neutral, but as you note, CO2 and other gases will dissolve in it, forming weak acids.
That happens when rain falls, but will also occur in the lab when distilled water is exposed to air. In that case, pH will be variably slightly less than 7.0. That's the reason calibration of pH meters relies on carefully prepared buffer solutions which are much less affected by airborne gases.
Neutrality of distilled water could be improved by boiling it to drive off atmospheric CO2, and storing it in full, sealed containers. If its pH were measured without exposure to air, it would be 7.0, or very close to it.
