I've never heard of a General Rodgers in Confederate service, which is not conclusive, but if there were such a man, he was pretty obscure. I am very well informed on this subject. "The first flag" is a meaningless phrase--there were several flags used by various forces in Confederate service. The one flag which might qualify was known as "the Bonnie Blue flag."
The flag on the left, the white star on the blue field, is "the bonnie blue flag." This sheet music is dedicated to Albert Pike. Albert Pike was a Massachusetts yankee who went west while still relatively young, and who eventually settled in Arkansas. He traveled extensively among the Indian tibes west of the Mississippi, especially in the Indian Territory (modern Oklahoma). Pike was commissioned a Brigadier General in Confederate service, and given command of the Indian Territory.