That seems unlikely to be a significant factor – gases in the atmosphere don’t show any significant sorting by weight, after all.
What, you don’t think people in the third world don’t wash their bottles too? (Come to think of it, I bet people in the first world are likelier, on balance, to just throw used bottles away.)
At equilibrium, sure. But when water vapor is added, the vapor buoyancy effect is significant in the movement of large air masses.
There is also evaporative cooling, so the air inside the bottle is slightly cooler and moister than the ambient air. What happens probably depends on ambient conditions, but with a long bottle and a small hole (something like a wine bottle), I suspect the position of the hole (top or bottom) may be significant.
Use a hair dryer, it’s fastest.
This sort of question comes up on photography forums all the time. People want to know what sort of container to keep their film and paper developing chemicals in (yes, they still make film, and some of us do not like the look of digital images), which is the best plastic for it, etc. It can get to be like how many angels can dance on the head of a pin.
If you don’t want drying marks on your jar/bottle/glass, give it a swirl of distilled water and put in on a towel face down w/ a knife or other utensil on one side to leave an air space. But really, the hair dryer is the way to go if you want to use it soon.