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We investigate the optical orientation, polarization pinning, and depolarization of optically confined semiconductor exciton-polariton condensates. We perform a complete mapping of the condensate polarization as a function of incident nonresonant excitation polarization and power. We utilize a ring-shaped excitation pattern to generate an exciton-induced potential that spatially confines polariton condensates into a single mode. We observe that formation of circular polarization in the condensate persists even for a weakly cocircularly polarized pump. By varying the excitation ring diameter we realize a transition from the condensate polarization being pinned along the coordinate-dependent cavity-strain axes, to a regime of zero degree of condensate polarization. Analysis through the driven-dissipative stochastic Gross-Pitaevskii equation reveals that this depolarization stems from a competition between sample induced in-plane polarization splitting and the condensate-reservoir overlap. An increase in the role of the latter results in weakening of the condensate fixed-point phase space attractors, and enhanced random phase space walk and appearance of limit cycle trajectories, reducing the degree of time-integrated polarization.