Methane isn't the most potent greenhouse gas as stated in the article. Water vapor is both the most potent and the most abundant. It also has a positive feedback loop so the hotter the Earth gets, the more water vapor goes into the atmosphere!
A reason to not include water vapor in a list of "greenhouse gases" is its short lifetime in the atmosphere. If you inject extra water vapor there, it rains down and exits the atmosphere in a matter of days, whereas methane etc stay for years --- humans emitting water vapor vs. CO2/etc to the atmosphere has very different relevance for the physics. Of course, the vapor does have effect on the radiation physics, and the fact that the equilibrium concentration depends on temperature leads to the positive feedbacks.
People leave out water vapor because, as you note, its concentration changes depending on temperature. It's an effect, not a cause. Emit lots of steam and you don't increase the temperature, you add to the rain.
Water vapor does amplify the climate effect of other greenhouse gases.
Methane is about 25-100 times more potent than CO2. Nitrous oxide about 300 times than CO2.
Although, the biggest worry is the stored methane. [1]
The amounts stored are equivalent (in CO2) to about 20-35 years of human activity (2014 CO2 output). If that gets quickly released, who knows what will happen.
I would add that presumably water vapour, methane and carbon dioxide have been in the atmosphere influencing temperatures since before anthropogenic climate change. I imagine rising concentrations of atmospheric water vapour are partly a consequence of emissions of carbon dioxide and methane emissions. (I believe that methane breaks down into chemicals including water.)
I imagine that humans are more responsible for levels of atmospheric water vapour indirectly (through carbon dioxide and methane emissions) than directly.
I imagine temperature and concentrations of atmospheric water vapour are in equilibrium in the absence of changes in the concentrations of other greenhouse gases.
Maybe I should have written my comment as questions. I would have phrased it differently if I had sources.
My main reason for answering the parent comment was to suggest that most of any increase in water in the atmosphere might largely result from warming due to other greenhouse gases. It might be the case that global warming is almost solely a function of greenhouse gases other than water vapour, even if it largely had its effect via water vapour.
The most potent greenhouse gas is actually SF6 (Sulfur hexafluoride), a chemical used in some industry processes. Water vapor is however the most common one.
SF6 is used as an insulator in high voltage breakers and switchgear. It is heavier than air so it sinks. There is a YouTube video where they float a paper boat on a basin of sf6 gas and also breathe it and it makes your voice low, opposite of helium
