It has nothing to do with on-demand, scheduled, domestic or any regulation, because there is no regulation that requires an empty flight to be Part 135 or Part 121 - therefore it is a Part 91 flight.
You don't have to dispatch an "empty" leg - that your company chooses to is to make it easier on the pilots and they presume safer in the operation (since that is the whole point of Part 135 and Part 121, to provide an additional layer of safety for the uneducated public).
I'm sorry - I've spent a lot of time working in this area, I know what I'm talking about. If you've just seen it from the pilot's point of view then it's easy to understand how you can be confused - because most companies that have millions of dollars tied up in aircraft decide that in the interests of safety they'll apply their operating certificate rules to empty legs. So to the pilot it looks like a Part 135 leg, smells like a Part 135 leg, and as far as they can tell is a Part 135 leg - but believe me when I tell you that to the FAA it's a Part 91 leg. They won't write you up for passing the FAF without visibility minimums, they won't write you up if you have exceeded the Part 135 duty and flight time maximums, they won't write you up if you haven't left a manifest at the departure etc. etc. They won't write you up because even if the pilot swears on the holy bible it's a Part 135 flight, they know the difference.