  We present exact solutions, mainly analytical, for the two-site
double-exchange-Holstein model, that allow us to draw a complete picture of the
role of both phonon and spin quantum fluctuations in determining the
short-range correlations in the manganites. We provide analytical solutions of
the model for arbitrary electron-phonon coupling and phonon frequency, for
$S=1/2$ and for the classical spin limit $S=\infty$, and compare these results
with numerical diagonalization of the realistic $S=3/2$ case. The comparison
reveals that the realistic case $S=3/2$ is not well described by the classical
spin limit, which is often used in literature. On the other hand, the phonon
fluctuations, parametrized by the phonon frequency $\omega_0$, stabilize
ferromagnetic phases with respect to the adiabatic limit. We also provide a
complete analysis on the polaron crossover in this model.
