The origin of Carbon Enhanced Metal-Poor (CEMP-no) stars with low abundances of neutron-capture elements is still unclear. These stars are ubiquitous, found primarily in the Milky Way halo and ultra-faint dwarf galaxies (UFDs). To make a major step forward, we developed a data-calibrated model for Böotes I that simultaneously includes all carbon sources: supernovae and asymptotic giant branch (AGB) stars both from first (Pop III) stars, and subsequent normal star formation (Pop II). We demonstrate that each of these sources leaves a specific chemical signature in the gas, allowing us to identify the origin of present day CEMP-no stars through their location in the A(C)-[Fe/H] diagram. The CEMP stars with A(C) > 6 are predominantly enriched by AGB Pop II stars. We identify a new class of moderate CEMP-s stars with A(C) ~7 and 0 < [Ba/Fe] < + 1, imprinted by winds from AGB stars. True Pop III descendants are predicted to have A(C) < 6 and a constant [C/Mg] with [Fe/H], in perfect agreement with observations in Böotes I and the Milky Way halo. For the first time we now have a complete picture of the origins of CEMP-no stars which can and will be verified with future observations.