Correct.
Mother is white. He is half black.
Raised by a white mother and her parents.

Everyone knows this. Being part black is still black. May not be full black, but part of him is still black.
If you have read his books you will know that he identifies himself as both African American and black. You would also know that people always treated him as an African American and he identified with that culture. So everyone else's opinions really don't matter.
Anyone arguing about his race or ethnicity are usually 1) black people who are resentful or have something against those who are part black, 2) biracial black people who want to separate themselves from other blacks & want nothing to do with black, 3) non-blacks who don't want to accept that their president is black or who try to suppress his identity to make themselves feel better.
Its all rooted in insecurity.