It should probably be both.  They are not inconsistent or alternative classifications.  You only get to be under section 509 if you are already classified under section 501(c)(3).  Section 509 subclassifies charities exempt under section 50s(c)(3).  Since you are considering 501(c)(3), I assume you have formed a charitable organization to pursue a charitable purpose.  An organization gains its charitable exemption by being recognized as an organization described in section 501(c)(3). 

But all charities are further divided, primarily between public charities and private foundations.  Public charities have a lot more freedom of operation than private foundations.   The distinction between public charities and private foundations is spelled out in section 509 of the Code. 

All charities are considered private foundations unless they meet one of the requirements set out in section 509.  Based on your description of your anticipated support being almost entirely from grants and contributions, you should probably be a public charity described in section 509(a)(1). (See Ready Reference Page: “Calculating Public Support Percentage”). 

If you are a charity, you are described in both section 501(c)(3) and section 509, with the distinction between public charity and private foundation being controlled by section 509.