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As you state, it should properly be called heat, since a process, though
not a uniquely specified one, is implicit. It is certainly not an energy.
Use of the symbol dU (by which I infer you mean the change in U) is most
improper; heat is not the change in anything. You should use Q instead:
Q = m c dT
Treating heat, work, and energy as though they are all the same thing
is [wrong].