Medieval knights did not, as a general rule, typically walk around in a white belt and plain chain, and the white belt and chain were not restricted to them alone. Try this simple test. Look at a dozen medieval depictions of knights from various periods, reproduced in color. Are they always wearing white belts? (No). Do they often wear belts of other colors? (Yes. In many periods it is very rare to find depictions of knights in white belts.) Do the knights wear simple chains, as opposed to the decorated chains of livery or of specific knightly orders? (Rarely, if ever) Now look for pictures of non-knightly types, peasants and simple infantry and the like. Can you find them wearing white belts? (Yes).
What did happen was this: as part of the ceremony of knighthood the candidate would be girded with a sword belt, a token of the profession of arms. Sometimes, but not always, the belt was white, to signify purity. The Ordene de Chevalerie, one of the few sources to specify white, also says that the belt should be small and narrow in token of humility!
But the evidence I mention above suggests that the white belt was not normally worn afterwards as a symbol of knightly rank. Think of it as being like a modern wedding dress: you wear white to the wedding but you don't spend the rest of your married life in a white dress to show people you are married.