Saruman would not have used any magical techniques taught by Aule when he was a Maia spirit dwelling in Aman. In fact, he would not have remembered such techniques, nor much of his actual existence in the Blessed Realm. Tolkien is quite explicit by saying that, when the Maiar spirits were incarnated into the bodies of the Istari wizards, they were henceforth limited in their powers, objectives, and even memories of their origins, as well as being subject to mortal fears, anxieties, weariness, etc., while obviously immune to so e of the ravages of time.
Moreover, the Istari all had to learn the ways of Middle-Earth Magic from scratch when they arrived—-through study, travel, wandering, etc. across the ages.
It seems to me that the reference to Saruman’s ring indicates nothing more than his moral collapse into a desperate attempt to fancy himself some manner of rival to Sauron (despite being almost completely under Sauron’s sway) and his frustration at not being able to find the One Ring he desired. Surely, Saruman’s skills would have allowed him to fashion some sort of magic ring completely on his own, with its own potencies, quite apart from the kind of power that went into the arts of Celebrimbor at Eregion.
Calling himself a Ringmaker was Saruman’s way of making himself feel on a par, somehow, with Sauron—-and let’s not forget, Saruman did indeed plan to “assist” Sauron’s reign (with Gandalf hopefully at his side) and eventually try to gain control and influence over him. He was deluded into thinking he could play the role of a rival Power to Sauron, hence the ring-making, the treachery, and the war-waging with his own, unique creations: the Uruks.