Other replies have explained that the ions and electrons can be at different temperatures, and that is of course true. I can offer some more explanation why. While you have linked temperature to some average kinetic energy, this is not a strictly accurate notion of temperature. Temperature only means something in the context of thermodynamic equilibrium; for any system out of equilibrium, often a temperature cannot be defined, even as an average kinetic energy. The two concepts are linked to each other, in the sense that an average kinetic energy at the particle scale is a significant (and sometimes only, but not always) component of the internal energy of the system. But temperature is first and foremost an equilibrium concept.

A plasma is composed of both ions (partially ionized atoms, or if they are fully ionized, bare nuclei) and electrons. Together, they constitute two co-located but separate fluids, whose motions may be distinct; indeed, because electrons are so much lighter than ions, they respond to forces much more readily. In equilibrium, a single fluid must have sufficiently rapid interactions so that the particle distributions are driven to a Maxwellian. In that case, for a single fluid, a temperature is well defined. In the case of a two-fluid system, self-interactions (such as interactions between ions and themselves, and electrons and themselves) may be sufficient to establish two separate equilibria corresponding to each fluid; hence, the electron vs. ion temperature. Only in the case that ion-electron interactions are sufficiently rapid would those equilibria be driven together to a single-temperature fluid, in which case, Te = Ti.

This picture becomes rapidly more complicated at very high temperature (usually, around 0.1 keV or about a million K), at which point the photons being exchanged by hot charged particles become dynamically important, and a third temperature, the radiation temperature, may emerge. At this point, the plasma must be described using three temperatures - if, and only if, you are in a situation lucky enough for equilibrium to manifest. (Fortunately, equilibrium is not usually that hard to access.) In many cases, such as when the system undergoes a strong shock, the system may be driven very far from equilibrium, but only temporarily. In others, some underlying energetic process may continuously drive the system away from equilibrium, resulting in a metastable state; this is the case in stellar atmospheres, in which NLTE (non-local thermodynamic equilibrium) processes matter a great deal. Usually, physicists resort to kinetic theory to try to understand such situations.