The pT-differential cross sections of prompt charm-strange baryons Ξ0c and Ξ+c were measured at midrapidity (|y|<0.5) in proton-proton (pp) collisions at a center-of-mass energy √s=13  TeV with the ALICE detector at the LHC. The Ξ0c baryon was reconstructed via both the semileptonic decay (Ξ−e+νe) 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(Ξ0c→Ξ−e+νe)/BR(Ξ0c→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 (pT) dependence of the Ξ0c- and Ξ+c-baryon production relative to the D0 meson and to the Σ0,+,++c- and Λ+c-baryon production are reported. The baryon-to-meson ratio increases toward low pT 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.