The p_{T}-differential cross sections of prompt charm-strange baryons Ξ_{c}^{0} and Ξ_{c}^{+} were measured at midrapidity (|y|<0.5) in proton-proton (pp) collisions at a center-of-mass energy sqrt[s]=13 TeV with the ALICE detector at the LHC. The Ξ_{c}^{0} baryon was reconstructed via both the semileptonic decay (Ξ^{-}e^{+}ν_{e}) and the hadronic decay (Ξ^{-}π^{+}) channels. The Ξ_{c}^{+} baryon was reconstructed via the hadronic decay (Ξ^{-}π^{+}π^{+}) channel. The branching-fraction ratio BR(Ξ_{c}^{0}→Ξ^{-}e^{+}ν_{e})/BR(Ξ_{c}^{0}→Ξ^{-}π^{+})=1.38±0.14(stat)±0.22(syst) was measured with a total uncertainty reduced by a factor of about 3 with respect to the current world average reported by the Particle Data Group. The transverse momentum (p_{T}) dependence of the Ξ_{c}^{0}- and Ξ_{c}^{+}-baryon production relative to the D^{0} meson and to the Σ_{c}^{0,+,++}- and Λ_{c}^{+}-baryon production are reported. The baryon-to-meson ratio increases toward low p_{T} up to a value of approximately 0.3. The measurements are compared with various models that take different hadronization mechanisms into consideration. The results provide stringent constraints to these theoretical calculations and additional evidence that different processes are involved in charm hadronization in electron-positron (e^{+}e^{-}) and hadronic collisions.