Rey's Falcon when did she gain ownership?

many characters, abilities, and scenarios are based on a "snap shot" according to the devs, so yes it is that moment when they had the ship leaving Jakku. Ownership is 9/10 of the law when it comes to illegal actions, but the dev team also stated that they base things like this on what the character would say, and at that point they would have likely said the ship was theirs.