  Superconductivity found in doped NdNiO$_2$ is puzzling as two local
symmetries of doped NiO$_2$ layers compete, with presumably far-reaching
implications for the involved mechanism: a cuprate-like regime with Zhang-Rice
singlets {\cblue is replaced by local triplet states at realistic values of
charge-transfer energy, which would suggest a rather different
superconductivity scenario from high-$T_c$ cuprates}. We address this
competition by investigating Ni$_4$O$_8$ clusters with periodic boundary
conditions in the parameter range relevant for the superconducting nickelates.
With increasing value of charge-transfer energy we observe upon hole doping the
expected crossover from the cuprate regime dominated by Zhang-Rice singlets to
the local triplet states. We find that smaller charge-transfer energy $\Delta$
is able to drive this change of the ground state character when realistic
values for nickel-oxygen repulsion $U_{dp}$ are taken into account. For large
values of the charge-transfer energy, oxygen orbitals are less important than
in superconducting cuprates as their spectral weight is found only at rather
high excitation energies. However, a second Ni($3d$) orbital can easily become
relevant, with either the $xy$ or the $3z^2-r^2$ orbitals contributing in
addition to the $x^2-y^2$ orbital {\cblue to the formation of triplet states.
In addition,} our result that $U_{dp}$ (acting between Ni and O) favors onsite
triplets implies that correlation effects beyond purely onsite interactions
should be taken into account when obtaining effective two-band models.
