It moved around even the OP went from talking about the empty leg (Par 91) to OpSpecs requirements (Part 135).

You're right though; the logging would be a little different since, according to the posts, the 135 leg would require who an SIC (per OpSepcs) while the Part 91 leg would not. That would mean differences in logging under 61.51(e)(1)(iii) and (f)(2) (ops requiring more than one pilot) although it wouldn't affect logging under (e)(1)(i) (the sole manipulator clause)