It has everything to do with how you present your seat availability. That's why many of the other attempts to sell seats on private jets involved memberships.
If you present seats for sale on your aircraft to anyone, you are, by definition, a 'common carrier'. Anyone can buy a ticket. The only question is, are you a 'scheduled' or 'unscheduled' airline.
While many corporate flight departments operate to Part 121 regs as a matter of their internal safety management system, most do not. It's VERY expensive.