What regulation allows a commercial pilot to receive compensation under part 91 when the other party supplies the plane?

Here's the scenario: I am a commercial pilot who is NOT holding out and am approached by a friend to fly her (for a fee) to Las Vegas in her small (less than 20 seats and sub 6,000lbs gross) plane. I've been told and internet research backs up the idea that this is OK without any need for an operating cert. However, if I supply the plane it requires an operating cert (this makes sense, 119.23 (b)).

Why is it OK under part 91? 91.501 appears to only apply to large, turbojet ME, and fractional arrangements. I've seen mention made of operational control being a factor, but cannot find anything official to back that up.