Yes I think too that this notation is a bit unfortunate (exactly as for your question on $\partial/\partial_{t_{coll}}$).

Maybe things are clearer if instead of writing:

$$\partial_tg = \partial_{t_0} g+ \epsilon^1 \partial_{t_1}g + \dots,$$

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 serie in power of $\epsilon$. Each terms in $(\dots)_i$ is of order $\epsilon^0$. BUT, they will contribute to 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 wavelenght since these are desribed 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 diffusive timescale ($w=Dk^2$). Somehow, for this reason it makes sense to use this weird multiple scale notation.