In the case I mentioned, a current is induced by moving a wire. In a transformer, it's the same process, but reversed. You move a magnetic field by inducing an alternating current in a wire. That field moves about another wire, inducing a current at a different voltage.
Yes, energy is stored, in the same sense that energy is "stored" when one places a heavy object on a high shelf. But there is nothing "magnetic" about the stored energy. Energy is energy. The energy associated with magnetism comes from the spin of electrons. I am sure you know all of this.
"Fully half the quotations found on the internet are either mis-attributed, or outright fabrications." - Abraham Lincoln