We investigate the influence of the accretion disk magnetic flux profile on the collimation of jets applying axisymmetric MHD simulations. We use the ZEUS-3D code modified for magnetic diffusivity. Our simulations evolve from an initial configuration in hydrostatic equilibrium in a force-free magnetic field. Considering a power law for the disk poloidal magnetic field profile B_P ~ r^-μ and for the disk wind density profile ρ ~ r, we perform a systematic study over a wide range of parameters μ and μ_ρ. We find the degree of collimation (quantified by the axial vs. lateral mass flux ratio) decreasing for steepening disk magnetic field profiles. Variation of the total magnetic flux does not affect the collimation degree substantially but changes the timescale of outflow evolution and the terminal jet speed. Our major result is a general relation between the collimation degree and the disk wind magnetization power-law exponent. Highly collimated outflows resulting from a flat disk magnetic field profile tend to be unsteady, producing axially propagating knots. Slightly depending on the disk wind density profile, this behavior sets in for μ < 0.4. We also perform simulations with artificially enhanced decay of the toroidal magnetic field component probing the idea of a "poloidal jet collimation" previously raised in the literature. These outflows remain weakly collimated and propagate slower. Our large numerical grid (7 × 14 AU for protostars) allows us to compare our simulations to observationally indicated jet rotation, suggesting a flat disk magnetic field profile, μ 0.5, in DG Tau. Our general results are applicable to both stellar and extragalactic sources of MHD jets.