Is $\kappa_e$ caused by charge-carrier transport or electron transport? $\kappa$ usually symbolizes thermal conductivity, a material's ability to conduct heat.
$\kappa$ can be expressed from its partial thermal conductivities - from the contributions of different phenomena:
$\kappa=\kappa_e+\kappa_{ph}+...$
where $\kappa_{ph}$ is from phonon transport and others may be appended.
My question is, what exactly is $\kappa_e$?
The usual description is: thermal conductivity from electron transport. But is it actually due to charge-carrier transport?
If so, then what about a p-type semiconducting material. Is $\kappa_e$ still electron transport or is it in this case rather hole transport since holes are the charge-carriers?
If so, this would flip around the direction of heat flow in such materials.
 A: As you have noted, there can be multiple contributions to the thermal conductivity.  Basically, any aspect of the material that can move 'independently' can transfer energy from one place to another. Since the electronic subsystem can often be taken as independent of the ionic (lattice) subsystem, those are the two main terms that are written out.  However, they are not the only ones.  For example, in certain systems you can have magnetic 'particles', magnetons, and those can carry energy around. Any subsystem that you can put energy into and that can move, can conduct heat.
Now, that being said, your last few sentences indicate that you have a conceptual issue with the electronic thermal conductivity in a semiconductor, or with the issue of holes vs electrons.  Holes are a very convenient concept. However, movement of holes is still the result of the movement of electrons. For every jump of a hole from here to there, an electron is jumping in the opposite direction.  We prefer to describe the motion of the one hole, rather than the collective motion of lots of electrons.
Next, the concept of heat conduction is totally unrelated to the charge (or even if there is a charge) of whatever is moving. The hot part of the material will have a thermal distribution of phonons/electrons/holes/magnetons/ions that relates to that temperature, and the cold part will have a different thermal distribution. Since the higher temperature material will have higher thermal energy particles, they will have higher velocities (e.g. electrons), or there will be more of them relative to the cold area (e.g. phonons).  Thus, the net flux through a plane between the two regions will be in favor of motion from hot to cold. The heat conduction really doesn't care exactly what is moving, it is the result of more movement on the hot side than the cold side.
Now, you may also get net electron (or hole) fluxes as well as a result of the thermal conduction.  This is the basis of thermoelectric devices that will generate electricity from a thermal gradient. But that is a different question... 
