  The metal-poor stars in the Galactic halo are thought to show the imprints of
the first (PopIII) stars, and thus provide a glance at the first episodes of
star formation. In this work, we aim at understanding whether all very
metal-poor stars formed in environments polluted by PopIII supernovae (SNe) and
at what level. With a general parametric model for early metal enrichment, we
study the chemical abundances (from C to Zn) of an environment imprinted by a
single PopIII SN. We investigate how these abundances depend on the initial
mass and internal mixing of PopIII stars, as well as on their SN explosion
energy. We then study how subsequent generations of normal (PopII) SNe affect
the PopIII chemical signatures. By comparing the observed chemical abundances
with our model predictions, we show that stars with [C/Fe]$>+2.5$ form in
environments polluted purely by low-energy PopIII SNe ($E_{\rm SN}<2\times
10^{51}$erg). At lower [C/Fe], stars can be imprinted either by PopIII only, or
also by normal PopII SNe. The probability of being enriched by PopII SNe
increases as [C/Fe] decreases. When PopII stars contribute more to the
pollution, they wash out the diverse chemical peculiarities left by the
different PopIII SNe, and the chemical dispersion between their descendants
decreases. We conclude that C-normal stars ($\rm [C/Fe] \leq +0.7$) have likely
been enriched by PopII SNe at a $\geq 50\%$ level and we identify in the
abundance scatter a key diagnostic to pinpoint the signature of PopIII SNe.
