Gary --
Many artillery commands on both sides functioned as infantry when necessary, and sometimes infantrymen served heavy artillery batteries.
That's not the criteria to be used for the War Department's desigation for a command.
When reorganized on March 1862, officers of the 1st Alabama Regiment were accepted in service as artillery. Pay for Colonel Steedman and subordinate field officers was higher than that received by their peers in the infantry arm of service.
Here's the relevant section of the Confederate law applying to pay for the different arms:
SEC. 15. The monthly pay of the colonel of the corps of artillery shall be two hundred and ten dollars; of a lieutenant-colonel, one hundred and eighty-five dollars; of a major, one hundred and fifty dollars, and when serving on ordnance duty, one hundred and sixty-two dollars; of a captain, one hundred and thirty dollars; of a first lieutenant, ninety dollars; of a second lieutenant, eighty dollars; and the adjutant shall receive, in addition to his pay as lieutenant, ten dollars per month. Officers of artillery serving in the light artillery, or performing ordnance duty, shall receive the same pay as officers of cavalry of the same grade.
SEC. 16. The monthly pay of the officers of the infantry shall be as follows: of a colonel, one hundred and ninety-five dollars; of a lieutenant-colonel, one hundred and seventy dollars; of a major, one hundred and fifty dollars; of a captain, one hundred and thirty dollars; of a first lieutenant, ninety dollars; of a second lieutenant, eighty dollars; the adjutant, in addition to his pay as lieutenant, ten dollars.
SEC. 17. The monthly pay of the officers of the cavalry shall be as follows: of a colonel, two hundred and ten dollars; of a lieutenant- colonel, one hundred and eighty-five dollars; a major, one hundred and sixty-two dollars; a captain, one hundred and forty dollars, a first lieutenant, one hundred dollars; a second lieutenant, ninety dollars; the adjutant, ten dollars per month, in addition to his pay as lieutenant.
The designation 1st Alabama Infantry is commonly seen in many otherwise reliable publications, such as Confederate Military History:
http://www.civilwarhome.com/1alainf.htm
The original 1st Alabama Regiment organized as infantry for a twelve-month term. Serving through 1861 manning the heavy guns and mortars at Pensacola, Fla., the regiment disbanded in mid-January of 1862. Seven of the original ten companies reorganized for three years or the war -- three new companies were assigned and the new regiment went on duty at Island Ten. Members of the original regiment who elected not to reenlist served out their twelve-month terms, the last of them being discharged at Island Ten.