Tom,

The version I've most frequently encountered is a combination of both you listed... ie: he received the nickname 'N****r Jack', in jest, from fellow graduates upon his being posted to the 10th; it stuck as a result of his long service with the Colored Cavalry; and the press subsequently sanitized it to 'Black Jack' in 1916 when he was chasing Villa around Mexico. I once met a Great War vet who referred to Pershing as "Old N***** Jack".

Regards,
Jim Ball