Visible crew/equipment: During the zero-gravity fight scene in the hotel room, it is possible to see a crew member's hand briefly resting against the end of the bed when Arthur and the projection are fighting on it.
Visible crew/equipment: Right before Cobb and Arthur go to the roof to find Saito having Nash in the Helicopter, Cobb and Arthur are in the Hotel Room with windows in the background. In the reflection of the windows, you can easily see the camera and the microphone.
Chosen answer: He bought out the actual airline company. If he'd bought out all the tickets for that specific journey he'd have said "I bought out the flight" or similar. It's a deliberately over the top moment of exuberance to highlight exactly how rich Satio is. I'm afraid that it is your friend who is correct, sorry.
Manky