Certainly pure, distilled, RO, tap, filtered, or any human drinkable water and over a wide range of pH will work and not matter in terms of germination rate. But distilled or other high purity water is not how things work in nature, and not having any nute/salt content in feed water may cause problems beyond initial germination. For example, some calcium and magnesium in the water, such as tap water left sitting for a day, would be good. So, Yes, you can feed "filtered tapwater" or any other consumable water to seeds and will see no difference in germination. The seeds deal with very wide variations in nature. But the seeds as soon as sprouted should be put in a low-dose nute-precharged medium or soil.
Otherwise, why the H2O2 pre-soak, exposing your seeds to a strong oxidizer? This is not how most do it, seed companies don't recommend this, and this is not how it happens in nature. As a "newbie" you should strive for simplicity, minimizing manipulations. Unless you have an actual good reason (what?), I suggest forget about H2O2 treating your seeds.
Why not just plant and sprout the seeds in their final pots/containers; and use the same water with your seeds as you'll be using to feed them later? If convenient, planting in their final container is probably the best option for a "newbie."