Water is the main component of interstellar ice mantles, is abundant in the solar system and is a crucial ingredient for life. The formation of this molecule in the interstellar medium cannot be explained by gas-phase chemistry only and its surface hydrogenation formation routes at low temperatures (O, O₂, O₃ channels) are still unclear and most likely incomplete. In a previous paper we discussed an unexpected zeroth-order H₂O production behavior in O₂ ice hydrogenation experiments compared to the first-order H₂CO and CH₃OH production behavior found in former studies on hydrogenation of CO ice. In this paper we experimentally investigate in detail how the structure of O₂ ice leads to this rare behavior in reaction order and production yield. In our experiments H atoms are added to a thick O₂ ice under fully controlled conditions, while the changes are followed by means of reflection absorption infrared spectroscopy (RAIRS). The H-atom penetration mechanism is systematically studied by varying the temperature, thickness and structure of the O₂ ice. We conclude that the competition between reaction and diffusion of the H atoms into the O₂ ice explains the unexpected H₂O and H₂O₂ formation behavior. In addition, we show that the proposed O₂ hydrogenation scheme is incomplete, suggesting that additional surface reactions should be considered. Indeed, the detection of newly formed O₃ in the ice upon H-atom exposure proves that the O₂ channel is not an isolated route. Furthermore, the addition of H₂ molecules is found not to have a measurable effect on the O₂ reaction channel.