How much water is the atmosphere losing to space?
It is not actual water what is lost to space, because in the high atmosphere water usually dissociate into other molecules or ions. The oxygen ion outflow is frequently assumed to be a proxy for the loss of water from the planetary atmosphere. In terms of global outflow rates for the Earth the rate varies from $10^{25}$ to $10^{26} s^{-1}$, depending on geomagnetic activity (reference).  

On the poster of the reference (sent to me by the author) we can read:


  If we assume oxygen corresponds to water loss (self-regulation, Hunten
  and McElroy [1970]), then an oxygen loss rate of ~$10^{25} s^{-1}$
  corresponds to ~300 $\text{g s}^{-1}$ of water loss. Over the age of the
  solar system (4.5 billion years ~ $1.4 \times 10^{17}$ s) this loss
  rate gives $4.2 \times 10^{19}$ g of water.


The current loss figure is equivalent ~25,920 liters per day, or 9,467 $\text{m}^3$ per year. And the reference of that figure seem to be the paper Escape of O+ through the distant tail plasma sheet, that used measurements from the STEREO‐B (Solar Terrestrial Relations Observatory) spacecraft.

That would correspond to a total loss over Earth's history of 42,000 $\text{km}^3$ of water, equivalent to about 12 cm of sea level change. However, that's a straightforward extrapolation of the current rate, because they acknowledge that they don't know how to model how Earth's magnetosphere would have behaved in the past when the Sun was weaker. Although, for a weaker Sun it would be reasonable expect smaller losses.

This value is radically different to that of the article you pointed to. In that case they use the loss of hydrogen instead of oxygen as proxy for water loss:


  Knowing how much hydrogen had disappeared from the oceans over the
  last four billion years enabled the researchers to calculate that the
  oceans have lost about a quarter of their water since the Earth’s
  early days.


They the say:


  Today the atmosphere is rich in oxygen, which reacts with both
  hydrogen and deuterium to recreate water, which falls back to the
  Earth's surface. So the vast bulk of the water on Earth is held in a
  closed system that prevents the planet from gradually drying out.


That suggests that the limiting factor for the existence of water is now the abundance atmospheric oxygen (although oxygen in rocks if very abundant). So it would make sense to use now Oxygen loss as proxy for water loss, but in the past, before the atmosphere was flooded with oxygen, maybe hydrogen loss was a better proxy.

They suggests that methanogenesis can free hydrogen atoms and make them liable to be loss to space, but when it is part of a water molecule it is safe. Methanogenesis would have been much more common in the past when the atmosphere was rich in methane (50 to 500 richer than today).

To wrap up, it seem that there is no consensus about the actual amount of water loss trough Earth's history. Different proxies give different values. Some proxies might be representative to some periods in Earth's past, but other proxies are more representative of other periods. And unfortunately we don't have yet a good reconstruction of the composition and density of the atmosphere throughout Earth's history.

But we do know with some confidence, that while you where reading this answer, one or two liters of water were loss to space.