Coming from the North of England I was brought up to believe that the phrase "Feed a cold, starve a fever" was indeed a misquotation, but not quite as your explanation goes. We would say "feed a cold, starve of fever!" It sounds the same if you say it quickly. In other words, if you behave so as to make a cold worse, you may catch a fever and starve: starve can mean feel cold or shiver, as in the expression: "you look half-starved, lad" when someone comes in out of the cold. So I conclude that the saying has nothing to do with food or eating, just good advice. Any thoughts?