With respect to the question, 'how it is that oxygenated water does not become hydrogen peroxide', one answer is that in natural sunlight waters it actual does create H2O2, albeit even then, in only small amounts.
More precisely, the presence of H2O2 and other reactive oxygen species (ROS) have been cited in the literature as occurring in illuminated natural waters (see, for example, ‘Reactive Oxygen Species in Natural Waters ‘, by Neil V. Blough and Richard G. Zepp). To quote from an online two-page preview :
The presence of ROS in aquatic ecosystems was first reported in 1966 by Van Baalen and Marler, who detected the presence of hydrogen peroxide (H202) in tens to hundreds of nanomolar concentrations in surface seawaters... Although their data were inadequate to specify the source of the H202, they postulated that photochemical reactions, biological processes, or atmospheric deposition might contribute to its presence in seawaters. Some years later, Swallow (1969) suggested that the hydrated electron ($\ce{e-(aq)}$) could be produced in seawater by the action of...photoionization of endogenous phenolic compounds. He further proposed that rapid reaction of $\ce{e-(aq)}$ with dioxygen (02) would yield superoxide ($\ce{.O2-}$), ultimately leading to the formation of H202 via disproportionation….
My updated understanding of these reactions is presented below which starts with a solvated electron, $\ce{e-(aq)}$ and $\ce{H+}$ per the reactions:
$\ce{O2 + e-(aq) ⇌ .O2- (aq)}$ (superoxide radical anion)
$\ce{H+ + .O2- (aq) ⇌ .HO2 (aq)}$ (hydroperoxyl radical with a pKa = 4.88)
$\ce{.HO2 (aq) + .HO2 (aq) -> H2O2 + O2 (aq)}$ Source: Equation (1)
In natural waters in sunlight, dissolved organic matter (DOM) can be a source of solvated electrons (see 'Photoproduction of hydrated electrons from natural organic solutes in aquatic environments') and electron holes:
$\ce{DOM + hv -> e- + h+}$
Also, the presence of transition metal ions and dioxygen via so-called metal auto-oxidation can be a potential source of superoxide:
$\ce{Fe(2+)/Cu(+) + O2 ⇌ Fe(3+)/Cu(2+) + .O2-}$ (Source example: Eq (Vl))
The cited work by Blough and Zepp further notes that since the 1980’s:
...environmental scientists have acquired evidence for the near-ubiquitous occurrence in surface waters of not only H202 and 02" but also singlet dioxygen..., the hydroxyl radical (OH), and organic peroxyl radicals (R02), as well as other transient intermediates that are either immediate precursors or products of the ROS. It is now recognized that the production of this diverse array of species is driven primarily by abiotic photochemical reactions involving naturally occurring organic (and sometimes inorganic) chromophores (Zepp, 1991; Zika, 1987; Zafiriou et al., 1984). Because of its high concentration in surface waters (-250 nanoM), O2 dominates the photophysics and photochemistry of these materials. The resulting reaction sequences are readily interpreted within the well established concepts of direct and sensitized photo-oxidations, potentially coupled to thermal autoxidation processes.