You bring up a great point. In the end, using the term "heat transfer" is a bit of a misnomer as heat is simply the transfer of energy across a temperature gradient. It would be more correct to refer to it as "Thermal energy transfer". However, as you can see in this link with the answer from Robert Reiland, its not totally correct either. ( Read here. ).

With respect to the term "total heat content," the semantics are the same; total thermal energy is probably a better way to refer to this. Mark Barton does a great job addressing this exact point. ( Read here ).