You are correct that that is not the scientific definition of sugar, which is generally any 5 or 6 carbon ring molecule.

However when most people talk about sugar they are referring to table sugar, which is a molecule of glucose (a six carbon ring sugar) chemically bonded to a molecule of fructose (a 5-carbon ring sugar) - otherwise known as sucrose. The first thing your body does when it sees sucrose is split them into a molecule of glucose and a molecule of fructose (this happens in your small intestine). Glucose can be metabolized right away, in any cell, however fructose needs a bit more processing in the liver before cells can use it to generate ATP.