I know it can be confusing, there were indeed Artillery units that did operate and man Air Defense weapons, but they were all lumped in with all Artillery units under the 1957 Combat Arms Regimental System or CARS, this would include Tube Artillery, both Towed and Self Propelled ( i.e. Armored) the only way one would be able to tell what type it was, was be the unit description or designator in parentheses at the end of the unit title. Plus soldiers who were selected (draftees) or volunteered (enlistees) for any one of those different types artillery would take their AIT at two different posts. The guys for Artillery that were to be Field Arty i.e. tube, and Ground to Ground Missile like the Sergeant Missile etc etc, went to Fort Sill for AIT, guys for Air Defense weapons, that's Ground to Air Missile, and Automatic Weapons, like the Quad 50 Caliber Machine Guns etc etc, went to Bliss for AIT.
Here are but five examples of Ground Arty for those days.
1st Battalion 14th Artillery (105mm Howitzer, Towed)
3th Battalion 16th Artillery (155mm Howitzer, Towed)
1st Battalion 27th Artillery (155mm Howitzer, Self Propelled)
4th Battalion 39th Artillery (8" Howitzer, Self Propelled)
1st Battalion 42nd Artillery (Honest John)
And of Air Arty.
1st Battalion 44th Artillery (Automatic Weapon, Self Propelled) the unit that operated those M44 Dusters.
2nd Battalion 44th Artillery (Pershing)
6th Battalion 56th Artillery (Hawk)
1st Battalion 65th Artillery (Nike Hercules)
During the 60s the Army's Artillery operated numerous Air Defense Battalions, but as you see, despite any local unit application of ADA on vehicle bumpers or unit stationary/paper work of some units that fired air defense weaponry, they were not called as we have seen Air Defense Artillery, the weapons were Missiles, to include Hawk missiles etc, the other type were called Automatic Weapons, to include Vulcan, Chaparral (first unit activated in May 1969, which by that time the Branch was already Split in to FA and ADA), Duster, and the old Quad .50 Caliber Machine Guns, also Searchlight units.
To reiterate. While the Artillery Branch was split in two in late 1968, the Artillery Battalions continued to be known under their 1957 CARS titles, it was in I think 1971 or 1972 ( not sure of the year there) that the Artillery units that operated Air Defense Weapons were finally designated Air Defense Artillery, as in the ADA, in example the unit that was a part of the 1st Cav Div when I was in it in 1980-81, the 1st Battalion 68th Air Defense Artillery,
To add one final fine point it would seem that those units under CARS, that operated Air Defense Artillery weapons tended, but not always, to be ones associated with the old Coast Artillery/ Anti Aircraft Artillery, here after 1957 they just were no longer officially called by those old titles.