When I was growing up in southern Oklahoma in the '40s and early '50s, I never saw any coal; never knew anyone who had any or used any. But, yep, I had to get out there and milk the cow by the light of a coal oil lantern. At least that's what everyone called it.
I don't remember just when common usage of the term "coal oil" changed to "kerosene", however . . . .
I do remember for sure that I sold both kerosene AND "white gas" (unleaded gas) for 10 cents a gallon in the late '50s in my Dad's Mobil service station.
The only use I can remember for the white gas was for those newfangled Coleman lanterns that were so much brighter, but more trouble, than our old kerosene lanterns. But as others have said, kerosene or coal oil, was used for lots of things.