[1] In theoretical thermodynamics, respected authors vary in their approaches to the definition of quantity of heat transferred.
One is from a primarily empirical viewpoint (which will here be referred to as the thermodynamic stream), to define heat transfer as occurring only by specified macroscopic mechanisms; loosely speaking, this approach is historically older.
The other (which will here be referred to as the mechanical stream) is from a primarily theoretical viewpoint, to define it as a residual quantity calculated after transfers of energy as macroscopic work, between two bodies or closed systems, have been determined for a process, so as to conform with the principle of conservation of energy or the first law of thermodynamics for closed systems; this approach grew in the twentieth century, though was partly manifest in the nineteenth.
These two concepts are coordinately coherent in the sense that they arise jointly in the description of experiments of transfer of energy as heat.
Axiomatic presentations of this stream of thinking vary slightly, but they intend to avoid the notions of heat and of temperature in their axioms.
As mentioned above, a diathermal wall may pass energy as heat by thermal conduction, but not the matter.
These things are important for the study of the Einstein coefficients, which relies partly on the notion of thermodynamic equilibrium.
In the presentation of Carathéodory, it is essential that the definition of the adiabatic wall should in no way depend upon the notions of heat or temperature.