There is still a trial for the Surname. "Aosta " represented the branch not really the Surname but the Tribunals will decide in the end.
Well, since 1948 Aosta is also a part of the surname, due to the constitutional provision that the predicates of the titles of nobility existing before 28 October 1922 shall serve as a part of the surname.
The trial (that should be finally decided in some time) has as its object the use of the surname "di Savoia". According to Vittorio Emanuele, who has started the trial, Amedeo and Aimone cannot call themselves "di Savoia" because their surname legally is "di Savoia-Aosta"; AFAIK he claims that their usage of the surname "di Savoia" is an injurious practice that violates Italian law (i.e. art. 6 of the Civil Code).
So Prince of Savoy is referring to the dynasty named Savoia, like Prince (Princess) of Orléans is referring to the dynasty named Orléans and like Prince (Princess) of Bourbon-Two Sicilies is referring to the House of Bourbon-Two Sicilies.
That's quite correct. Basically Prince of Savoy is the title that all the dynasts have, according to the 1890 Royal Decree; in addition the children of the King and of the eldest son of the King are also Royal Princes of Italy.
The Royal Princes are styled Royal Highnesses; the others (who were called Princess of the Blood in the Decree) are styled Serene Highnesses, although during the Monarchy basically all them were upgraded to Royal Highnesses.
A Prince of Savoy can have a territorial designation like Duke of Aosta, a Prince of Orléans can have a territorial designation like Duke of Vendôme and a Prince of Bourbon-Two Sicilies can have a territorial designation like Duke of Castro.
Rather than a territorial designation, they are additional titles. In the case of the title "Duke of Aosta", "Duke" is the title and "Aosta" is the territorial designation of the title.
IIRC the Royal Decree about the titles provided that the sons of the King and of the Hereditary Royal Prince shall be given an additional title. Such titles in some cases were hereditary (i.e. the titles of Duke of Aosta and of Duke of Genova), in other cases were conferred
ad personam (i.e. the title of Duke of Spoleto conferred to the late Prince Aimone, or the titles of Duke of Bergamo, of Pistoia and of Ancona conferred to the younger sons of the late Prince Tommaso, Duke of Genova).