Integrating proteins in molecular electronic devices requires control over their solid-state electronic transport behavior. Unlike "traditional" electron transfer (ET) measurements of proteins that involve liquid environments and a redox cycle, no redox cofactor is needed for solid-state electron transport (ETp) across the protein. Here we show the fundamental difference between these two approaches by macroscopic area measurements, which allow measuring ETp temperature dependence down to cryogenic temperatures, via cytochrome C (Cyt C), an ET protein with a heme (Fe-porphyrin) prosthetic group as a redox centre. We compare the ETp to electrochemical ET measurements, and do so also for the protein without the Fe (with metal-free porphyrin) and without porphyrin. As removing the porphyrin irreversibly alters the protein's conformation, we repeat these measurements with human serum albumin (HSA), 'doped' (by non-covalent binding) with a single hemin equivalent, i.e., these natural and artificial proteins share a common prosthetic group. ETp via Cyt C and HSA-hemin are very similar in terms of current magnitude and temperature dependence, which suggests similar ETp mechanisms via these two systems, thermally activated hopping (with ~0.1 eV activation energy) >190 K and tunneling by superexchange <190 K. Also, ET rates to and from the Fe redox centres (Fe(2+) <=> Fe(3+) + e(-)), measured by electrochemistry of HSA-hemin are only 4 times lower than those for Cyt C. However, while removing the Fe redox centre from the porphyrin ring markedly affects the ET rate, it hardly changes the ETp currents through these proteins, while removing the macrocycle (from HSA, which retains its conformation) significantly reduces ETp efficiency. These results show that solid-state ETp across proteins does not require the presence of a redox cofactor, and that while for ET the Fe ion is the main electron mediator, for ETp the porphyrin ring has this function.