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:
$$\partial_t = \partial_{t_0} + \epsilon \partial_{t_1} + \dots$$
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 ($w=Dk^2$).
