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We propose two effective parameters that fully characterise galactic-scale structure formation at high redshifts ($z\gtrsim5$) for a variety of dark matter (DM) models that have a primordial cutoff in the matter power spectrum. Our description is within the recently proposed ETHOS framework and includes standard thermal Warm DM (WDM) and models with dark acoustic oscillations (DAOs). To define and explore this parameter space, we use high-redshift zoom-in simulations that cover a wide range of non-linear scales from those where DM should behave as CDM ($k\sim10\,h\,{\rm Mpc}^{-1}$), down to those characterised by the onset of galaxy formation ($k\sim500\,h\,{\rm Mpc}^{-1}$). We show that the two physically motivated parameters $h_{\rm peak}$ and $k_{\rm peak}$, the amplitude and scale of the first DAO peak, respectively, are sufficient to parametrize the linear matter power spectrum and classify the DM models as belonging to effective non-linear structure formation regions. These are defined by their relative departure from Cold DM ($k_{\rm peak}\rightarrow\infty$) and WDM ($h_{\rm peak}=0$) according to the non-linear matter power spectrum and halo mass function. We identify a region where the DAOs still leave a distinct signature from WDM down to $z=5$, while a large part of the DAO parameter space is shown to be degenerate with WDM. Our framework can then be used to seamlessly connect a broad class of particle DM models to their structure formation properties at high redshift without the need of additional $N$-body simulations.