We investigate anew the distribution of absolute carbon abundance, A(C)=log{epsilon}(C), for carbon-enhanced metal-poor (CEMP) stars in the halo of the Milky Way, based on high-resolution spectroscopic data for a total sample of 305 CEMP stars. The sample includes 147 CEMP-s (and CEMP-r/s) stars, 127 CEMP-no stars, and 31 CEMP stars that are unclassified, based on the currently employed [Ba/Fe] criterion. We confirm previous claims that the distribution of A(C) for CEMP stars is (at least) bimodal, with newly determined peaks centered on A(C)=7.96 (the high-C region) and A(C)=6.28 (the low-C region). A very high fraction of CEMP-s (and CEMP-r/s) stars belongs to the high-C region, while the great majority of CEMP-no stars resides in the low-C region. However, there exists complexity in the morphology of the A(C)-[Fe/H] space for the CEMP-no stars, a first indication that more than one class of first-generation stellar progenitors may be required to account for their observed abundances. The two groups of CEMP-no stars we identify exhibit clearly different locations in the A(Na)-A(C) and A(Mg)-A(C) spaces, also suggesting multiple progenitors. The clear distinction in A(C) between the CEMP-s (and CEMP-r/s) stars and the CEMP-no stars appears to be as successful, and likely more astrophysically fundamental, for the separation of these sub-classes as the previously recommended criterion based on [Ba/Fe] (and [Ba/Eu]) abundance ratios. This result opens the window for its application to present and future large-scale low- and medium-resolution spectroscopic surveys.