Moisture is a byproduct of the combustion cycle since there is water in gasoline. If you're only running for 20 minutes every few months, that stays in the oil and doesn't burn off completely. You don't need that living in the sump for 2 years. What are you trying to save here? the cost of a quart of oil or your generator? Just change the dang oil more often and don't worry about whether or not your opened bottle of oil will go bad on the shelf (it won't). Oil is cheap. I bet your generator isn't.
Water is a byproduct of the combustion, because that's what it turns into from hydrogen and oxygen, not any water already in the gas. It's around 32% of the combustion byproducts, though particularly with ethanol blends, a much smaller % of exhaust may be water from that too.
If you run for 20 minutes, the oil is getting up to temperature and vaporizing off the moisture. If you change the oil prematurely, it still gets into the oil the next time you run the engine, and vaporizes off again. Unless you intend to change the oil after every time the engine runs, it doesn't make as much difference as you imply.
What are you hoping to improve here? Doing something with no real benefit is not just about trying to save the cost of oil. There are an unlimited number of ways I don't senselessly waste my time and money. I've never had an engine fail from using oil that sat around. Has anyone? Almost exclusively it is instead due to insufficient oil changes based on hours ran or extreme duty, cooking the oil at high temp.
As far as "proper storage", oil is designed to handle around 250F operating temp for its life cycle, so does it really matter if stored at 65F vs 95F? Doubtful. The oil in my garage sees down to about 20F in winter and 100F in summer, for upwards of a decade. I have not had any oil related engine problems, including equipment and vehicles over 20 y/o. At the same time, I recognize that air cooled engines tend to heat and degrade the oil faster, cannot run same # of hours per oil change. The manufacturer almost always specifies the OCI, though when they state X hrs or yearly, I ignore the yearly part unless it is also not too far off X hrs too, then mainly for convenience of seasonal outdoor equipment, I'll change it at end of season.