What Debye length should be used in plasma? 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.
 A: I looked into a few plasma physics text books [1,2], and according to them the Debye length is defined as 
$$
\lambda_{D,j} = \left( \frac{\epsilon_0 k_B T_j}{n_j e^2} \right)^{\frac{1}{2}},
$$
where the index $j$ denotes either electrons or ions. We thus have an electron Debye length, and an ion Debye length. 
In addition, the total Debye length is defined as
$$
\frac{1}{\lambda_D^2} = \frac{1}{\lambda_{D,e}^2} + \frac{1}{\lambda_{D,i}^2},
$$
which is the same as you have written it, just arranged in a different way.
Let's calculate a few examples assuming $n_e=n_i$:
\begin{array}{ccc|ccc}
\hline
T_e \mbox{ in eV} & T_i \mbox{ in eV} & n_{e,i} \mbox{ in}\ \mathrm{m}^{-3} & \lambda_{D,e} \mbox{ in m} & \lambda_{D,i} \mbox{ in m} & \lambda_{D} \mbox{ in m}\\ \hline
1  & 1  & 10^{17}  &  2.3\cdot10^{-5} & 2.3\cdot10^{-5}  &  1.6\cdot10^{-5}\\
10 & 10 & 10^{17}  &  7.2\cdot10^{-5} & 7.2\cdot10^{-5}  &  5.1\cdot10^{-5}\\
10 & 1  & 10^{17}  &  7.2\cdot10^{-5} & 2.3\cdot10^{-5}  &  2.2\cdot10^{-5}\\ \hline
\end{array}
As you have said in your question, for $T_e \gg T_i$ the ion Debye length will dominate or, in other words, the total Debye length will be closer to the ion Debye length. 
In the laboratory, however, $T_e$ is often easier to measure than $T_i$. This is of course not generally true, but using for example Langmuir probes is a simple diagnostic to get the electron temperature. My answer is therefore that the electron Debye length is often used because the electron Temperature is relatively easy to measure (compared to the ion temperature). 
If the plasma density is higher, like in fusion plasmas, coupling between ions and electrons is strong and one can assume $T_i\approx T_e$, for which the total Debye length is close to the electron (or ion) Debye length. For such cases it is a valid approximation to use the electron Debye length instead of the total Debye length. 
In summary, the most important thing is to state explicitly what you are using (electron/ion/total Debye length). Since the difference between these quantities is usually not too large (especially when you just want to compare the Debye length to the plasma dimension to check on the validity of the plasma definition), often the electron Debye length is used. 

[1] A. Dinklage et al.: Plasma Physics: Confinement, Transport and Collective Effects (Springer, 2005)
[2] A. Piel: Plasma Physics: An Introduction to Laboratory, Space, and Fusion Plasmas (Springer, 2010)
