I wanted a clarification of this designation (notwithstanding the fact that the title is not necessarily used in all kingdoms): has it traditionally been reserved only for an heir-apparent who is a son of a king? What if he is a grandson of a king, whose father (the previous crown prince) has predeceased him?
The heir apparent to a king or emperor is always a crown prince, even if that is not the formal style or title used.
If a prince is born second and direct in the line of succession, and his father dies, does he become the new crown prince? I believe that the title Hereditary Prince is used in some kingdoms, to denote a direct heir is not first in line to the throne ...
Has he not moved up to being direct heir and first in line then? He would then become crown prince.
Of course, it is also used to denote the heir-apparent to a principality (e.g. Monaco and Liechtenstein). It goes without saying that the title Crown Prince is never used in those places -- or even in grand duchies (e.g. Luxembourg).
Right, the title of crown prince is only for the heirs of kings and emperors, it is misapplied in any other situation.
Can "Crown Prince" actually be used for an heir-presumptive who is first in line to the throne -- with the understanding that the said person can lose that designation?
Possibly, but it seems it has only been an approximation for other titles in the native language. For instance Diadochos in Greece... which really means “heir” or “successor”. I am not sure if that’s actually a title or a description though.