Heat is internal energy in motion, that is flowing from a hot region to a cold region. Once the energy arrives, it is no longer heat; it is internal energy. Heat and internal energy are not the same thing.
Work Step by Step
A similar relationship between heat and internal energy exists between work and KE/PE. Work is done on an object, giving it KE/PE, but the object does not "contain" any work. This is discussed on page 287.