Thermal Energy Transfer and Equilibrium

Internal energy $U$ is what a system holds; heating $Q$ is energy crossing a boundary because of a temperature difference. The transfer runs from higher temperature to lower temperature, whatever the totals are, because the exchange happens in collisions between individual particles and temperature is the average energy of one particle. Thermal equilibrium is the state of equal temperature, at which the net transfer is zero while particles keep exchanging energy in both directions at matching rates. Conduction, convection and radiation are three routes for the same transfer.

Three errors dominate. Speaking of the heat an object contains, or of cold moving into a warm object, which invents a second fluid and makes the first law impossible to write down correctly. Choosing the direction of transfer by asking which system holds more thermal energy, which sends energy out of a large cool pool and into a small hot nail. And reading equilibrium as an interaction that has stopped rather than one that has balanced, which leaves no way to explain why a room-temperature object still radiates.