The confinement of ultracold neutrons (UCNs) in three-dimensional magnetic field gradient or magneto-gravitational traps allows for a measurement of the free neutron lifetime τ_n with superior control over loss channels related to UCNs interacting with material surfaces. The most precise measurement τ_n has been achieved using a magneto-gravitational trap, in which UCN are prevented from escaping at the top of their trap by gravity. More compact horizontal confinement geometries with variable energy acceptance ranges can be obtained by using steep magnetic field gradients in all spatial directions, generated by combinations of either permanent or (variable) superconducting magnets. In this paper, we present the first successful implementation of a pulsed spin-flip based loading scheme to fill a three-dimensional magnetic trap with externally produced UCN. The measurements with the τSPECT experiment were performed at the pulsed UCN source of the research reactor TRIGA Mainz. The extracted neutron storage time constant of τ = 859(16) s is compatible with the most precise determinations of τ_n . We report on detailed, but statistically limited, investigations of major systematic effects influencing the neutron storage time. The statistical limitations are mitigated by the relocation of the experiment to a stronger UCN source.