Spin transport is crucial for future spintronic devices operating at bandwidths up to the terahertz range. In F|N thin-film stacks made of a ferromagnetic/ferrimagnetic layer F and a normal-metal layer N, spin transport is mediated by (1) spin-polarized conduction electrons and/or (2) torque between electron spins. To identify a crossover from (1) to (2), we study laser-driven spin currents in F|Pt stacks where F consists of model materials with different degrees of electrical conductivity. For the magnetic insulators yttrium iron garnet, gadolinium iron garnet (GIG) and γ−Fe2O3, identical dynamics is observed. It arises from the terahertz interfacial spin Seebeck effect (SSE), is fully determined by the relaxation of the electrons in the metal layer, and provides a rough estimate of the spin-mixing conductance of the GIG/Pt and γ−Fe2O3/Pt interfaces. Remarkably, in the half-metallic ferrimagnet Fe3O4 (magnetite), our measurements reveal two spin-current components with opposite direction. The slower, positive component exhibits SSE dynamics and is assigned to torque-type magnon excitation of the A- and B-spin sublattices of Fe3O4. The faster, negative component arises from the pyrospintronic effect and can consistently be assigned to ultrafast demagnetization of minority-spin hopping electrons. This observation supports the magneto-electronic model of Fe3O4. In general, our results provide a route to the contact-free separation of torque- and conduction-electron-mediated spin currents.