As mentioned above, a diathermal wall may pass energy as heat by thermal conduction, but not the matter. A diathermal wall can move and thus be a  part of a transfer of energy as work. Amongst walls that are impermeable to matter, diathermal and adiabatic walls are contraries.
For radiation, some further comments may be useful.
In classical thermodynamics, one-way radiation, from one system to another, is not considered. Two-way radiation between two systems is one of the two mechanisms of transfer of energy as heat. It may occur across a vacuum, with the two systems separated from the intervening vacuum by walls that are permeable only to radiation; such an arrangement fits the definition of a diathermal wall. The balance of radiative transfer is transfer of heat.
In thermodynamics, it is not necessary that the radiative transfer of heat be of pure black-body radiation, nor of incoherent radiation. Of course black-body radiation is incoherent. Thus laser radiation counts in thermodynamics as a one-way component of two-way radiation that is heat transfer. Also, by the [Helmholtz reciprocity] principle, the target system radiates into the laser source system, though of course relatively weakly compared with the laser light. According to Planck, an incoherent monochromatic beam of light transfers entropy and has a temperature. For a transfer to qualify as work, it must be reversible in the surroundings, for example in the concept of a reversible work reservoir. Laser light is not reversible in the surroundings and is therefore a component of transfer of energy as heat, not work.
In radiative transfer theory, one-way radiation is considered. For investigation of Kirchhoff's law of thermal radiation the notions of absorptivity and emissivity are necessary, and they rest on the idea of one-way radiation. These things are important for the study of the Einstein coefficients, which relies partly on the notion of thermodynamic equilibrium.
For the thermodynamic stream of thinking, the notion of empirical temperature is coordinately presupposed in the notion of heat transfer for the definition of an adiabatic wall.
For the mechanical stream of thinking, the exact way in which the walls are defined is important.
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. This is achieved by careful wording and reference to transfer of energy only as work. Buchdahl is careful in the same way. Nevertheless, Carathéodory explicitly postulates the existence of walls that are permeable only to heat, that is to say impermeable to work and to matter, but still permeable to energy in some unspecified way; they are called diathermal walls. One might be forgiven for inferring from this that heat is energy in transfer across walls permeable only to heat, and that such are admitted to exist unlabeled as postulated primitives.
The mechanical stream of thinking thus regards the adiabatic enclosure's property of not allowing the transfer of heat across itself as a deduction from the Carathéodory axioms of thermodynamics, and regards transfer as heat as a residual rather than a primary concept.