In short
Which expression should be used: (1) or (2)?
At length
Definition of the Debye length $r_D$ is following:
$$r_D=\left( \sum_{j}\frac{4\pi q_j^2 n_j}{kT_j} \right)^{-\frac{1}{2}}$$
It is calculated from the Poisson's equation in electrostatics with help of the Boltzmann distribution for every $j$-th charge species.
Let's consider two component plasma, which is consisted of electrons with charge $e$ and ions with charge $Ze$. Then the Debye screening radius looks like so:
$$r_D = \left[ \frac{4\pi e^2}{k} \left( \frac{n_e}{T_e} + \frac{Z^2 n_i}{T_i} \right) \right]^{-\frac{1}{2}} = \left[ \frac{k T_e}{4\pi e^2 n_e \left( 1 + Z \frac{T_e}{T_i} \right)} \right]^{\frac{1}{2}}$$
In experiments we often have that electrons temeprature $T_e$ is significantly larger than ions temperature $T_i$. It means that the Debye distance is determined only by ions:
$$r_D = \left( \frac{k T_i}{4\pi e^2 n_e Z} \right)^{\frac{1}{2}} = \left( \frac{k T_i}{4\pi (Ze)^2 n_i} \right)^{\frac{1}{2}} \tag{1}$$
But nevertheless people use another form in their theoretical and experimental investigations:
$$r_D = \left( \frac{k T_e}{4\pi e^2 n_e} \right)^{\frac{1}{2}} \tag{2}$$
despite $T_e \gg T_i$. And that means that electrons play crucial role in screening, and ions don't. I understand this fact only qualitative: electrons are lighter and more mobile than ions, and only they can quickly compensate any charge excess. But I don't understand how to prove that using (2) is accurate and using (1) is not.
Wikipedia says the following:
Even in quasineutral cold plasma, where ion contribution virtually seems to be larger due to lower ion temperature, the ion term is actually often dropped, although this is only valid when the mobility of ions is negligible compared to the process's timescale.
and gives the link on the Hutchinson's book. But I can't find there any proof of the statement.