Background
Generally speaking, in plasma physics one uses the terms cold and hot to refer to the ratio of the thermal, $V_{T,j}$, to bulk flow speeds, $V_{o,j}$. Meaning, a cold(hot) plasma has $V_{T,j} \ll V_{o,j}$($V_{T,j} \gg V_{o,j}$). A quick reference of the possible thermal speeds can be found in this answer. For more information on cold vs. hot, see this answer.
For a proton-electron plasma, if $T_{e} \approx T_{p}$, then due to $m_{e} \ll m_{p}$ one can see that $V_{T,e} \gg V_{T,p}$.
To think about temperature in a plasma you need to be careful about the type of plasma. In a plasma that is not controlled by Coulomb collisions, the temperature is not as clearly defined as one might recall from fluid dynamics. One can still calculate the various velocity moments of the distribution function, but temperature becomes more of a measure of the average kinetic energy of a species in its bulk flow rest frame than the standard thermodynamic quantity.
Questions
Question number one: can plasmas generated by heat, like stars, only be thermal, or can they be non-thermal too?
Generally speaking, for a plasma to be considered "thermal" in the context of which you speak, the gas would need to be dominated/controlled by Coulomb collisions. The result would be a Maxwellian-like velocity distribution (i.e., gaussian).
The lower atmosphere (i.e., low chromosphere and photosphere) of the sun is thought to be a collisional plasma, thus much of the plasma is thermalized. However, once the plasma reaches the upper atmosphere (i.e., corona), Coulomb collisions start to lose relevance to other faster mechanisms like wave-particle interactions. This is all ignoring the complications introduced by transient phenomena like solar flares and coronal mass ejections.
The solar wind is generally considered to be weakly collisional at best, but for most scenarios it is usually assumed to be collisionless. The velocity distributions of both the ions and electrons are typically composed of several components.
Ions
For the ions, they are often composed of the following components:
- the core/beam of the bulk flow, which is a cold bi-Maxwellian;
- a secondary proton beam streaming along the quasi-static magnetic field; and
- an alpha-particle beam.
Electrons
For the electrons, they are often composed of the following components:
- a core distribution which has $V_{T,ec} \gg V_{o,ic}$ ($c$ stands for core) but generally follows the ion core in bulk flow and can often be modeled as a bi-Maxwellian;
- a hotter halo is not Maxwellian-like, it is better modeled by a kappa distribution (which is similar to having power-law tails extend off of the core);
- a strahl component, which is an anisotropic (in temperature) beam streaming along the quasi-static magnetic field.
Answer
So both the ions and electrons have multiple components, many of which are not modeled as thermalized distributions (i.e., non-Maxwellian profiles). Thus, stars are fully capable of generating what many people would call nonthermal plasmas.
Question number two: then, what causes the two temperatures to be that different in laboratory plasmas? What's the difference with fusion ones, why have they $T_{i} \approx T_{e}$?
This depends upon how the plasmas were made and what the relevant parameters (e.g., plasma beta, ratio of cyclotron-to-plasma frequency, etc.). It may also depend upon the type of ions used, since the mass ratio between ions and electrons is relevant for energy exchange processes between the two species.
The reasons relate to how the plasma is heated once formed. One way of doing this in fusion research is to use the instabilities that naturally occur within all plasmas. An instability is a mechanism by which free energy can be transferred to/from the electromagnetic fields from/to the particles. Generally, the free energy exists in the particles as a non-Maxwellian feature (e.g., a secondary beam or temperature anisotropy) and the instability then radiates an electromagnetic wave. During the process of the radiation, the energy used to produce the wave is taken out of the particle distributions in an attempt to remove the free energy. Once the wave exists, it can then further interact with the plasma (interesting side note: the wave interacts most strongly and efficiently with the types of particle distributions that radiated them in the first place).
Some of these instabilities are very good at transferring energy between the ions and electrons (e.g., lower-hybrid drift, ion-acoustic, whistler-types, etc.). These types of instabilities can lead to situations where $T_{i} \approx T_{e}$ (However, as I stated above, if the plasma is not controlled by collisions, then the temperature is really defined by component.). There is something to note though. The waves radiated by instabilities will interact with particles over a specific range of energies and phases, not the entire distribution. Thus, they often lead to further non-Maxwellian features rather than relaxing the distribution to an isotropic Maxwellian.
Answer
So to answer your question, I am inclined to think the temperature equilibriation in fusion plasmas results from the instabilities generated (and sometimes used) to heat the plasma. The plasmas in any lab device are generally created for a specific purpose, so the experimenters generally want specific parameters for various levels of control. Unfortunately, there are dozens of reasons why $T_{i} \approx T_{e}$ and why $T_{i} \neq T_{e}$ in plasmas. Without knowing further specifics, I cannot further elaborate.
