I would call Jamaica Farewell a West Indian ballad rather than a calypso. This is partly because it is not Trinidadian, partly because the melody is not one of the common calypsos.

Many of the songs that Belafonte performed were not calypsos, even though they were marketed as such. One of the major conflations was between calypso and mento, which was a Jamaican musical form that pre-dated ska. "Jackass Song" is an example of a mento performed by Belafonte, later made into ska by Prince Buster.

I find Belafonte's studio recordings bland, but his live work is incredible and I can't recommend too highly the 1959 live double album recorded at the Carnegie Hall.