Bose-Einstein condensates of exciton-polaritons are known for their fascinating coherent and polarization properties. The spin state of the condensate is reflected in polarization of the exciton-polariton emission, with temporal fluctuations of this polarization being, in general, capable of reflecting quantum statistics of polaritons in the condensate. To study the polarization properties of optically trapped polariton condensates, we take advantage of the spin noise spectroscopy technique. The ratio between the noise of ellipticity of the condensate emission and its polarization plane rotation noise is found to be dependent, in a nontrivial way, on the intensity of continuous wave nonresonant laser pumping. We show that the interplay between the ellipticity and the rotation noise can be explained in terms of the competition between the self-induced Larmor precession of the condensate pseudospin and the static polarization anisotropy of the microcavity.