Radiatively generated, LO quark (u,d,s,c,b) and gluon densities in the real, unpolarized photon, improved in respect to our previous paper F. Cornet, P. Jankowski, M. Krawczyk and A. Lorca, Phys. Rev. D68, 014010 (2003), are presented. We perform three global fits to the F₂\gamma data, using the LO DGLAP evolution equation. We improve the treatment of the strong coupling running and use lower values of  \Lambda _QCD, as we have found that the too high values adopted in the previous work caused the high \chi ² of the fits. In addition to the modified FFNS_CJKL model, referred to as FFNS_CJK1 we analyse a FFNS_CJK2 model in which we take into account the resolved-photon heavy-quark contribution. New CJK model with an improved high-x behaviour of the F₂^\gamma \left (x,Q²\right ) is proposed. Finally, in the case of the CJK model we abandon the valence sum rule imposed on the VMD input densities. New fits give \chi ² per degree of freedom about 0.25 better than the old results. All features of the CJKL model, such as the realistic heavy-quark distributions, good description of the LEP data on the Q² dependence of the F₂\gamma and on F_2,c\gamma are preserved. Moreover we present results of an analysis of the uncertainties of the CJK parton distributions due to the experimental errors. It is based on the Hessian method used for the proton and very recently applied for the photon by one of us. Parton and structure function parametrizations of the best fits in both FFNS_CJK and CJK approaches are made accessible. For the CJK model we provide also sets of test parametrizations which allow for calculation of uncertainties of any physical value depending on the real photon parton densities.

PACS numbers: 13.60.Hb, 14.70.Bh, 14.65.Dw, 14.65.Fy