Three different derivations of the classical binary nucleation theory are considered in detail. It is shown that the derivation originally presented by Wilemski [J. Chem. Phys. 80, 1370 (1984)] is consistent with more extensive derivations [Oxtoby and Kashchiev, J. Chem. Phys. 100, 7665 (1994)]; Debenedetti, Metastable Liquids: Concepts and Principles (Princeton University Press, Princeton, 1996) if and only if the assumption is made that the surface of tension of the binary nucleus coincides with the dividing surface specified by the surface condition Sigma n(si)v(li)=0, where the n(si) denote surface excess numbers of molecules of species i, and the v's are partial molecular volumes. From this condition, it follows that (1) the surface tension is curvature independent and (2) that the nucleus volume is V=Sigma n(li)v(li)=Sigma g(i)v(li), where the n(li) are the numbers of molecules in the uniform liquid phase of the droplet model encompassed by the surface of tension, and the g(i) are the total molecular occupation numbers contained by the nucleus. We show, furthermore, that the above surface condition leads to explicit formulas for the surface excess numbers n(si) in the nucleus. Computations for the ethanol-water system show that the surface number for water molecules (n(s,H2O)) causes the negative occupation numbers (g(H2O)) obtained earlier using the classical nucleation theory. The unphysical behavior produced by the classical theory for surface active systems is thus a direct consequence of the assumption of curvature independence of surface tension. Based on the explicit formulas for n(si), we calculate the full free-energy surfaces for binary nucleation in the revised classical theory and compare these with the free-energy surfaces in the Doyle (unrevised classical) theory. Significant differences in nucleus size and composition are found between these models and they are related to surface excess density. It is shown that only for the revised classical theory is the nucleus composition consistent with the Gibbs dividing surface model.