  We contrast {\it four} distinct versions of the BCS-Bose statistical
crossover theory according to the form assumed for the electron-number equation
that accompanies the BCS gap equation. The four versions correspond to
explicitly accounting for two-hole-(2h) as well as two-electron-(2e) Cooper
pairs (CPs), or both in equal proportions, or only either kind. This follows
from a recent generalization of the Bose-Einstein condensation (GBEC)
statistical theory that includes not boson-boson interactions but rather 2e-
and also (without loss of generality) 2h-CPs interacting with unpaired
electrons and holes in a single-band model that is easily converted into a
two-band model. The GBEC theory is essentially an extension of the
Friedberg-T.D. Lee 1989 BEC theory of superconductors that excludes 2h-CPs. It
can thus recover, when the numbers of 2h- and 2e-CPs in both BE-condensed and
noncondensed states are separately equal, the BCS gap equation for all
temperatures and couplings as well as the zero-temperature BCS
(rigorous-upper-bound) condensation energy for all couplings. But ignoring
either 2h- {\it or} 2e-CPs it can do neither. In particular, only {\it half}
the BCS condensation energy is obtained in the two crossover versions ignoring
either kind of CPs. We show how critical temperatures $T_{c}$ from the original
BCS-Bose crossover theory in 2D require unphysically large couplings for the
Cooper/BCS model interaction to differ significantly from the $T_{c}$s of
ordinary BCS theory (where the number equation is substituted by the assumption
that the chemical potential equals the Fermi energy).
