My reference is J.A. Bittencourt's book Fundamentals of Plasma Physics.
Cold plasma
One understands under a cold plasma a particular choice of closure for the infinite series of moments of the Boltzmann equation.
In the momentum equation appears the pressure dyad (energy density). In a cold plasma this term is taken to be zero (no limit implied) and hence the series is truncated. As a result such a plasma fluid model only consists of mass and momentum equations.
This means however, that there is no thermal motion of particles (e.g. particle flux due to temperature gradient) or forces due to the divergence of the kinetic pressure tensor. But there are still collisions for momentum transfer.
In fact it follows that the plasma temperature is zero and the velocity distribution function is a Dirac $\delta$-function centered at the macroscopic
flow velocity $u(r,t)$ $$f_\alpha(r,v,t)=\delta(v-u(r,t))$$
Warm plasma
As a side note the next easier closure is the heat flux vector in the energy density equation to be taken zero. This is also called the adiabatic approximation.
Collisionless plasma
Now, the term collisionless plasma is studied in relation to propagation of electromagnetic waves in the plasma. In this context the collision frequency $\nu$ appears in the dielectric constant of the medium. Here, collisionless usually means that the collision frequency is much less than the wave frequency $\omega$. $$\nu \ll\omega$$. If this holds, the resulting dispersion relations can be significantly simplified.
However, to properly study the propagation of electromagnetic waves in the plasma, one has to have a model for the medium first, meaning one has to pick a closure relation for the momemt equations.
E.g., as another side note, the theory for wave propagation in a cold plasma (as defined above) is known as the magnetoionic theory.
As a result one has dispersion relations for a cold plasma with and without collisions as well as for a warm plasma and so on.