Yes, I think too that this notation is a bit unfortunate (exactly as for your question on $\partial/\partial_{t_{\text{coll}}}$).
Maybe things would be clearer if, instead of writing:
$$\partial_t g = \partial_{t_0} g + \epsilon \partial_{t_1} g + \dots$$
we write:
$$\partial_t g = (\partial_t g)_0 + \epsilon ^1(\partial_{t} g)_1 + \dots,$$
where we assumed that the result of a time derivative can just be expanded into a series in power of $\epsilon$. Each terms in $(\dots)_i$ is of order $\epsilon^0$. But they will contribute to a different physical process. The zeroth order term will give you the Euler equation; hence, they correspond to phenomena happening on a time scale similar to the inverse wavelength since these are described by waves or advection ($w=ck$). The first-order term will give you the dissipative contributions (viscosity, heat diffusion, ...) to Navier-Stokes, which happen on a diffusive timescale ($w=Dk^2$).
This somehow justifies the use of derivative with different "timescales", as in regular multiscale perturbation. Indeed, $\epsilon$ keeps track of the gradients. The zeroth order correspond to term proportional to single gradient and the first order to term proportional to double gradient (Lalacian) in the transport equations. Hence $\partial_{t_0}$ corresponds to the change in time of some variable due to the single gradients terms, $\partial_{t_1}$ corresponds to the change in time due to double gradients.. and so on