Seeking name of a small natural gas generator that will run in all weather
IMHO all of these 'don't run in the rain' rules mean, "don't let it get wet"

So the easy answer is to build a small shed for it, or locate it under sufficient overhang.

I have a gas generator with the same proviso -- don't operate in rain.  I took a barrel, split it in half verticaly, and the generator wears it like a hat.