Question

One person having the authority, granted by the courts, to make decisions for another person is called

Answer 1

I believe it's called Guardianship
Guardianship could took form in either limited authority to make decisions for the decision (where the other person still could took part)  or full authority where the guardian have the full right to make the decision.
Guardianship could only be applied if the defendant is a minor or unable to functioned properly.