Correct, contracting the Flu is the result of infection by some sort of pathogen in our bodies, not explicitly by being cold or wet. However, having our internal body temperature reduced, even by a couple of degrees, can compromise our immune systems to the extent that a dormant Flu-inducing bacteria or virus can take hold. Which is why the sick-day cliche often involves a character getting cold/wet prior as to create a sense of cause and effect, and tie plot points together like in this case.