Pure distilled or RO water has no pH of it's own and is perfectly fine to give straight to your plants at any time. It will not affect the pH in the pots or reservoir either way. pH pens won't generally read pure water correctly. Leave your pen in either if you want to destroy it.
When a pure water is exposed to air it absorbs CO2 creating a week carbonic acid sol'n that may read as low as 5.5 but the acidity level is so weak that a drop of pH Up will make it slightly alkaline, (over pH7). That's why it won't affect the pH in the root zone by itself.
I use nothing but those forms of water and have for the last 20 years. My diploma in chemistry makes me sure of my facts.