We theoretically investigate a spinor polariton condensate under nonresonant pumping, based on driven-dissipative Gross-Pitaevskii equations coupled to the rate equation of a spin-unpolarized reservoir. We find the homogeneous polariton condensate can transit from the spin-unpolarized phase, where it is linearly polarized, to the spin-polarized phase, where it is elliptically polarized, depending on the cross-spin versus same-spin interactions and the linear polarization splitting. In both phases, we study elementary excitations using Bogoliubov approach, in a regime where the decay rate of total exciton density in reservoir crosses over from the slow to the fast limit. Depending on reservoir parameters, the global-phase mode can be either diffusive or gapped. By contrast, the relative-phase mode always possesses a gapped energy, undamped in the spin-unpolarized phase but weakly damped in the spin-polarized phase. In the spin-unpolarized phase, both modes are linearly polarized despite pumping and decay. However, in the spin-polarized phase, the mode polarization can be significantly affected by the reservoir and depends strongly on the circular polarization degree of the condensate. Interestingly, we demonstrate that the 'ghost' branch of the Bogoliubov spectrum of the relative-phase mode can be visualized in the photoluminescence emission, distinguishable from that of the global-phase mode and thus allowing for experimental observation, when the spinor polariton condensate is elliptically polarized.