John,
It might be the same battle, but the accounts I have make it sound like a totally different battle.

Round Hill is 10 miles from Curtis' camp on Cache Bayou and 12 miles from Round Hill.

From the diary pages it says:
On the 7th of July the 13th Illinois broke camp at 2 A.M. marched 20 miles from Augusta to Peach Orchard Bluff and camped. Had a sharp battle. Enemy loss was 17 killed, 83 wounded. Regiment loss 12 killed & 30 wounded. The battle was fought mainly by sharpshooters.
On the 8th of July the 13th Illinois broke camp at 7 A.M. and marched 10 miles to Cache Bayou and camped in the woods.

Curtis' army camped at Augusta on the 4th, 5th and 6th of July. They marched from Augusta to Cache Bayou and camped in the woods. They had the battle on the 7th of July at Round Hill. On the 8th of July Curtis' army camped on the
east side of Bayou De View. Estimated casualties: 308 total (US 63; CS 245)

I understand that the northern troops were strangers to this area and they often labeled geographical features wrong, but the casualty list from the battles should be very close to the same as the other Union troops that were there. Also the distance traveled from Augusta to Peach Orchard Bluff on the 7th then on the 8th to Cache Bayou would suggest another engagement and not Hill's Plantation.
I will keep digging and hopefully I will be able to find out if this is the same engagement or a totally different engagement.

Thanks for the insight though.