We analyzed the functional superconducting characteristics of MgB₂-based bulks without and with additions of Ti, Ti-O and TiC, prepared by hot pressing (30 MPa), spark plasma sintering (50-96 MPa) and under high quasy-hydrostatic pressure (2 GPa) conditions. Their stability in gas hydrogen under 4.2 bar pressure was under the study. The trapped magnetic fields were studied using hollow cylinders of the same geometry prepared from magnesium diboride-based materials and MT-YBCO. The high critical current densities and critical magnetic fields should ensure high trapped fields in all these materials. Indeed all materials demonstrated the required performance; however, flux jumps are a serious issue in MgB₂ even in crack free cylinders and impeded higher trapped fields. An inhomogeneous and porous MgB₂ structure was found to be less stable against flux jumps. On the other hand, deviations of the material matrices from MgB₂ stoichiometry did not impede high J_c and trapped fields. The superconducting properties of all materials investigated in this study occurred to be sufficient for magnets in submersible liquid hydrogen pumps with a required trapped field of about 500-600 mT.