In a plasma, the electron temperatures and ion temperatures, when they exist, generally refer to a kinetic temperature as would be applied in a Maxwell-Boltzmann distribution for the velocities, just as the OP specifies. The temperature therefore means the same thing in both cases, although for a given temperature, the corresponding velocity distributions are different in each case because the masses of the particles may be different.
Also as the OP notes, particles in a plasma may not be thermalized, and so their velocity distributions in those cases would not be well-represented by a Maxwell-Boltzmann distribution. In principle this can be true for either the electrons, the positive ions, or both. It is also possible for the electrons and ions to be thermalized at separate temperatures.
It's also the case that the velocities of the electrons and positive ions are coupled in a variety of ways in a plasma, the simplest being two-body coulomb collisions. If the time scale for these collisions to take place is sufficiently short compared to other time scales at which the system is evolving, then the electrons and positive ions will equilibrate at a common temperature. If, however, this timescale is too long, then the two particles are not likely to equilibrate at a common temperature.
But again, the temperature is used in the same way for both collections of particles: to specify the velocity distribution when it is thermalized.