A bit more evidence that your ancestor was in the 6th Alabama Cavalry:

There was a J A White in Company B (one card shows Co. H), of the 6th Alabama Cavalry. He is listed on the surrender documents as being a resident of Opelika, Russell County. For much of his service he was detailed as a "forage master", for which he received an extra 25 cents per day. However, there is one additional card in his file that appears to be misfiled. It shows that J A White, in a detachment from "Co. I, 6th Ala. Cav. Gen. Clanton's Brig. at Coosa Bridge" appears on a receipt roll for clothing received Oct. 6, 1864. This separate individual appears to be your ancestor, and this card may be the only direct record of his service, aside from the statements of witnesses on his pension application.

His headstone was not ordered until 1930. It seems pretty clear that when the stone was ordered, as when the census was taken, the branch "Cavalry" was left off the paperwork.