What Rick Tarara suggests should be done, I'm afraid, is undoubtedly
what will be done, but I don't think that students are well served by
it. I think this is especially true in the cases where the revisionist
version is either too complicated to be understood or else just plain
wrong.
You may call the quantity m*c*dT "thermal energy" if you want to, and
if that were the only impropriety around I wouldn't object so strongly.
However with "heat flow" and other blatant heresies about I'm afraid
the beautiful simplicity of the theory is lost. Will you similarly call
m*L thermal energy? (L is a latent heat per unit mass associated with
some phase transition.) The transition may be brought on by heating
(pardon me - warming) the system, but no temperature rise is seen. Does
a different kind of energy get transferred in this process? What about
the common process of taking a cube of ice and melting it on a room
temperature surface? We certainly give those students this problem at
entry level. Are they to be told that heating the ice adds vibrational
thermal energy, melting the ice overcomes binding energy, and further
heating of the water increases kinetic thermal energy? I tell them all
that is conceptually necessary to understand what is happening in the
system without gratuitous introduction of those extra terms. There is
only one "kind" of energy.
The only possible reason I can imagine for discriminating among some
arbitrarily defined kinds of energy is so that one can later ask
students on an examination to parrot what has been taught to them. It's
not physics; physicists don't use those terms. In my opinion it
shouldn't be allowed to clutter a subject which is already somewhat
bewildering to many students.