Hydrogen is very, very abundant, so the creation of water is always going to be a thing as long as there is oxygen. We also know that the period referred to as

Hydrogen is very, very abundant, so the creation of water is always going to be a thing as long as there is oxygen. We also know that the period referred to as the Late Heavy Bombardment is likely to have showered the already formed Earth with water-rich ice from the outer solar system. Most of this water would be older than the sun. The article only mentions the Great Oxygenation Event, but it's worth noting that biological organisms today are breaking down and creating water all the time as an essential part of their metabolism. It's not inconceivable that a sizeable portion of water was originally synthesised out of rocky compounds and atmospheric gases (such as methane) by archaic organisms. All in all, we have identified many different sources for water on Earth, signs point to all of them being a factor. We're just not sure how large the individual factors are, but many people consider ice from the Late Heavy Bombardment to be the biggest one.