Who pays is only one test for whether an operation is 135 or private.
There are others including "holding out", "common purpose", etc. I
would say your example sounds more like 135 than private. Also, be
aware the the FAA has said that, in general, the ability to log time is
considered compensation. Some groups (such as Civil Air Patrol and some
glider groups) have specific wavers from the FAA for this.

-Robert, CFI