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The metal-poor stars in the Galactic halo are thought to show the imprints of the first (Pop III) 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 Pop III 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 Pop III SN. We investigate how these abundances depend on the initial mass and internal mixing of Pop III stars, as well as on their SN explosion energy. We then study how subsequent generations of normal (Pop II) SNe affect the Pop III 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 Pop III SNe (E SN < 2 × 1051 erg). At lower [C/Fe], stars can be imprinted either by Pop III only, or also by normal Pop II SNe. The probability of being enriched by Pop II SNe increases as [C/Fe] decreases. When Pop II stars contribute more to the pollution, they wash out the diverse chemical peculiarities left by the different Pop III SNe, and the chemical dispersion between their descendants decreases. We conclude that C-normal stars (⁠|$\rm [C/Fe] \le +0.7$|⁠) have likely been enriched by Pop II SNe at a |$\ge 50~{{\ \rm per\ cent}}$| level and we identify in the abundance scatter a key diagnostic to pinpoint the signature of Pop III SNe. [ABSTRACT FROM AUTHOR]