::Actually, , it's not.  The "key description as 'privately funded'" is pretty accurate in that the first hundred million bucks of development, and the creation of the company was all Elon Musk, and another hundred million was from SpaceX VC investors.  In addition to the c. $200 million private funding, some specific development $$$ were paid by NASA for specific cargo vehicle characteristics that NASA wanted for cargo vehicles to take cargo to the ISS, and later about $75 million (to date) for specific development of crew carrying capabilities that NASA wanted SpaceX to develop.  Most of the other $$$ paid to SpaceX by NASA are merely the payments to date for buying two launches of a medium-class launch vehicle (these typically cost >$100 million per launch, although SpaceX prices to NASA are, I believe, quite a bit lower (per pound of payload) than the United Launch Alliance sells launches of its' flagship launch vehicles, the Atlas V and the Delta IV), or for partial payments for future cargo launches (12 of them under contract) where, due to US accounting rules on long-lead-time capital-intensive products, it has been usual for years in (even the commercial satellite) launch industry to pay substantial progress payments to the manufacturer beginning two years prior to the nominal launch date.  So SpaceX may have some hundred-millions of NASA money, but by far, most of it is merely payments for specific launch services, either already flown, or with launch vehicles in production, and a much smaller part of it is government-paid costs for SpaceX to develop specific product characteristics that the government wanted.   

 ::So you can't really count the purchase of specific SpaceX launches by NASA as government funding of the development of SpaceX's launch vehicles, the Falcon 1 and the Falcon 9.  This is all documented in a great many sources.  Hope this is helpful to you.  Cheers.