And of course there was the election of 1824, where Jackson had more popular votes, and more electoral votes, than John Quincy Adams, but the number of non-whites voting would have been very small and not a factor in the outcome.
You forgot about Hayes in 1876—he clearly got a *lot* fewer white votes than did Tilden (blacks voted 95%+ Republican back then).
You are correct about 1880 and 1888:
Garfield barely beat out Hancock in the popular vote (thanks to Hancock getting a supermajority of the white Southern vote), so he definitely got beat out among white voters.
Harrison lost the popular vote to Cleveland in 1888, so he definitely lost the white vote by a large percentage.
Republican winners from 1896-1928 won the popular vote by quite a bit, and non-whites were a smallish percentage of the vote, so the winner of the white vote won every time.
In 1948, I believe that Truman carried the white vote. Truman carried the popular vote over Dewey by 49.55% to 45.07%, and blacks voters were neither so numerous (due to segregation) nor so universally Democrat (while many black voters switched to the Democrats with the New Deal, a notable percentage of them stuck with the GOP) that Dewey could have made up that 4.5% deficit.
In 1960, Kennedy barely won the popular vote, and won good margins among blacks, Native Americans, and non-white Hispanics (not certain who carried the Asian vote, but it was neither large nor one-sided enough to make a difference), so he definitely lost the white vote. LBJ, though, won in a landslide and carried the white vote handily (albeit not in the South).
Carter in 1976 won the popular vote by only 2%, and blacks gave him a huge percentage of the vote (85%), so I don’t believe he could have carried whites. Clinton must have lost the white vote in 92 and 96 even with Perot outperforming among white voters, and Obama in 2008 got 53% overall but far less than that among white voters.
So the winner of the white vote won the election except for Gilded Age Republicans and post-Brown v. Board Democrats (with LBJ as the only counterexample).