Is there more oxygen in Earth’s water (seas, oceans etc., including oxygen in water molecules) or in the atmosphere?
Earth’s water masses at 1.35 x 10^18 tonnes. Approximately eight ninths of that is oxygen, so 1.2 x 10^18 tonnes.
The oxygen in the atmosphere masses at 1.2 x 10^9 tonnes. So there is approximately 1 000 000 000 times as much oxygen in water molecules as there is in the atmosphere.
Given this estimate, the oxygen dissolved in water can probably be discounted as irrelevant.
According to the EMD Periodic Table app, the earth’s core is 49.4% Oxygen, so maybe there is even more under our very feet?
The core is primarily iron. Are you sure they aren’t talking out the the crust with it silicon and aluminium oxides?
I’ll be damned they did get it wrong - http://pse.merck.de/merck.php?lang=EN
O - Percentage mass of the earth’s core - 49.4%
Fe - Percentage mass of the earth’s core - 4.7%
ETS - sent a message to the company, let’s see what they say. 
Assuming they say anything.
A German company, eh?
Weren’t Hitler/the Nazis proponents of the hollow Earth theory? Couldn’t a hollow Earth have those proportions of elements in its “core”? Wasn’t Hitler’s body never conclusively found?
Hmmm…
ETA: I’m not saying there’s Nazis living inside the hollow Earth, but… Nazis.
Maybe the core is just really rusty?
OP: In a can on the shelf over the refrigerator, behind the coffee.
Well, let’s throw this out there, then.
In the past, the O2 level in the atmosphere has been up to 50% or so higher than now. Where’d that O2 come from and where’d it go? Less dissolved in the oceans? Released from melting ice? Hollowing out of the core? What?
It came from the oxygen in water, split off by plants and photosynthetic bacteria.
I don’t think the OP realises quite how big the oceans are, and how thin and tenuous the earth’s atmosphere is.
I believe the higher amounts of oxygen (and CO2) in the past have gone into rock formation.
The size discrepancy isn’t as obvious as you seem to think. Water covers 75% of the earth, air covers 100% of the earth. The deepest water on earth is roughtly 6 miles below sea level, and the atmosphere, depending on how you measure, extends 60 miles up, or maybe a hundred miles, or more.
I’m not disputing that most of the oxygen is in the water, but that’s not particularly related to size/volume. It’s more related to composition.
Some rough (probably inaccurate) math suggests that the mass of oxygen in the atmosphere in metric tons is around 1.05E+15, where as the mass of just the water on earth is around 1.35E+24: if one counts the oxygen bound in the water molecules, which accounts, I think, for 7/8s of the water’s mass (one O is about 16 times “heavier” than one H), that would put the raw oxygen mass of the earth’s oceans at somewhere in the neighborhood of 1.2E+24 metric tons. Although, I suspect this is not what the OP is looking for.
This page says that the peak concentration of O[sub]2[/sub] in sea water is 20 parts per million, under the most ideal circumstances. If we were to guess that the average is one tenth of that (2ppm), using basic math, one comes up with 4.76E+18 metric tons of oxygen in the sea, or around five thousand times as much as in the atmosphere. I suspect my math is off by a bit, though, because it ignores chemistry and other things.
I’m guessing they meant “Percentage mass of the earth”, not “percentage mass of the earth’s core”.
But then again, this IS a .de domain. Who knows if the people who wrote the webpage even speak English?
Think of it in terms of pressure. It only takes about 10 meters of water to get a full atmosphere’s worth of pressure, and the oceans are, on average, much more than 10 meters deep.
The Wikipedia page listing the abundance of elements in the crust gives similar numbers. The crust is ever so slightly closer to us and a tad easier to study, presumably the core’s numbers are derived from that plus seismic analysis.
The weight of the atmosphere on you is similar to being at the bottom of a 10-foot deep pool. That’s differential of course, you hardly notice it.
But to answer your question, the vast bulk of oxygen is still locked in the crust, mantle and core.
Need answer fast?