Zeroth Order Approximation
In the simplest approximation, cold implies that $T_{e} = T_{i} = 0$, where $T_{s}$ is the average temperature of species $s$. There is an entire branch of plasma theory based upon this assumption. It is another way of saying that you assume the plasma is initially at rest with no thermal fluctuations. It also implies the plasma has no pressure, thus no pressure waves can exist if $T_{e} = T_{i} = 0$.
I wrote an answer describing potential wave modes in such a system at: https://physics.stackexchange.com/a/138460/59023.
Finite Temperature Approximation
In a slightly less extreme approximation, one can argue the plasma is cold when the plasma beta, $\beta$, is very small. That is:
$$
\beta = \frac{2 \mu_{o} \ n_{o} \ k_{B} \left( T_{e} + T_{i} \right) }{B_{o}^{2}} \ll 1
$$
where $n_{o}$ is the charged particle number density, $B_{o}$ is the quasi-static magnetic field, $\mu_{o}$ is the permeability of free space, and $k_{B}$ is the Boltzmann constant. I wrote an answer describing how to define the particle temperatures at: https://physics.stackexchange.com/a/218643/59023.
Phenomenological Answer
The answer to your question really depends upon the application or circumstances in which you are interested. For instance, we have found through observations that the whistler mode wave, or the R mode when $\Omega_{ci} < \omega < \Omega_{ce}$ (where $\Omega_{cs}$ is the cyclotron frequency of species $s$), is well characterized by cold plasma dispersion relation in the high density limit (i.e., $\omega^{2} \ll \omega_{pe}^{2}$ and $\Omega_{ce}^{2} \ll \omega_{pe}^{2}$, where $\omega_{ps}$ is the plasma frequency of species $s$) even though we know the plasma is not cold.
It is another way of saying that there are circumstances where the temperature corrections do not have a noticeable impact on the system/phenomena in which you are interested.