Sam Manning first recorded "Hold Him Joe (My Donkey Wants Water)" in 1925 for Columbia Records (2409-X). Columbia credits the song to Manning himself but I'm quite sure it was already a well known Jamaican "digging song".
Here's the original 78... oh how I'd love a copy!!!
http://www.popsike.com/SAM-MANNING-Hold-Him-Joe-COLUMBIA-2409X-E/360312746382.html

As for Lord Executor, he did not record anything in 1910. It is a well known fact that the first "Calypso" (Calinder music with vocals) on record was Iron Duke in 1914. Executor came to fame for his battles with WIlmoth Houdini in the '30s (ie; "My Reply to Houdini"). I've never heard of him even performing "Hol' Him Joe".

Belafonte barely wrote anything. All of his best songs were covers of well known Calypsos. His "Jackass Song" sounds like a cover of "Donkey Brae" (recorded in 1956 by Lord Flea) but even that was taken from part of a 1951 Mento-Calypsos medley by Lord Fly.
Yu nuh hear weh de old man seh? Yu no tie no donkey dung deh. Jackass fi jump and brae, mek him brae, mek him brae.