The important difference is that temperature is a measurement quantity that can be inserted in a formula (or recipe), while heat is a relative term, not a scientific term, and is, as the questioner suggests, the energy in a substance or process, caused by, residing in, or sustained by the movement of molecules. Anything with a temperature high than absolute zero (-274 Kelvin) has some thermodynamic energy (the scientific term for heat). The confusion lies in the terms used for the physical state, as opposed to the man-made measurement of that energy. Heat is also the name for one kind of energy, and that energy can be transformed to other kinds: kinetic, electrical, etc. “Heat” is also a member of the general English vocabulary, usually paired in the duality “heat-cold”, not exactly polar opposites, but applicable in context. The terms may be applied at many temperatures: “cold tea”, “cold weather” “hot tea”, hot weather”, and as a metaphor: “cold case” “hot date”, etc.; temperature, however, cannot.