Azizi, tha list has several that could be argued about with good points on both sides. I think the jury is still out on some of my picks.
With a nod to Michael Morris, lullabies have common themes found in many countries. They also tend to borrow verses (example- "Black Sheep Lullaby," which seems to be black, and "All the Pretty Little Horses." verse which goes a ways back in both B and W.
Old Fiddle tunes- "Cotton-eye Joe" for one- seem to have an old white tune at the base. Many floating verses from many songs. "Cotton-eye" seems to be a white expression for a person with prominent white in the eyeball, or so the language books tell me.
"Frankie and Johnny" is a remake, B, of "Frankie and Albert," whose first writing could be B or W, and based on a still older story, W.
"Oh Mary" is black, but there are possibly some white 'floaters."
Shortnin' Bread- B
Grizzly Bear- the "Preacher and the Bear," W, but variants both B and W.
Note: The grizzly bear never ranged into eastern or southern United States.
John Henry- sung by everybody. Related to John Hardy? No many different verses (???). African-Americans have made it their own, but origin?
Take This Hammer- collected from B, and I think it originated with B.
Boll Weevil- Texas, White, about 1900, but immediately taken up by both B and W since it was a real disaster for both. The composers are named in some books, but I haven't seen proof.
Michael Rowed- B
Midnight Special- B
Nine Pound Hammer- ? I think earliest mentions are B
Old Aunt Kate and Old Aunt Katie- both old fiddle tunes. W origin.
Or is there a song by that name unrelated to the fiddle tunes?
Polly Woddle- A Harvard parody (ca. 1880) of a minstrel tune. The tune is a fiddle tune. W
Railroad Bill- seems to be two stories here, one B, one W. Take a raincheck on this one.
Run, Children, Run- is this R N R? B modified by W in recordings.
Paw Paw Patch- All Georgia white kids know this one. I imagine all black kids too (or they would have when I was a kid). I would guess W but wouldn't bet my savings on it.
Grey Goose- several. But if it is the one sung by Lead Belly, it's B.
Mister Rabbit- B, but first put in print about 1880 by a white author, Joel Chandler Harris. He preserved much of the black storyteller's art in his books.
Bile dem Cabbage. Another fiddle tune. Nearly all floating verses. Who wrote the cabbage verse, B or W, I won't try to guess.

A challenging list.