Not a drop of truth to that chestnut urban legend. Onions are safe no matter if they are cut or not and whether stored refrigerated or not, safe. Just think of the many times you have cut then stored that cut onion in the cooler. Have you ever been sick from eating it later, no. One reason is that onions produce sulphuric acid. This is only done once the onion is cut when enzymes within the onion act on it to make the acid as a defense against consumption. The acid keeps the onions pH low enough to prevent any contaminating organisms from proliferation.
*Always be aware of causes of cross contamination in your food prep areas.