The internal behavior is undocumented, and varies with the operating system.
We have evidence that copyfile() on linux invokes the system "cp" command to do the copying. It then becomes plausible that on linux, movefile() would invoke linux's "mv" command. The behaviour would then be determined by the OS utility. In Linux, "mv" on the same filesystem changes directory entries but does not make a copy, and the overhead would be very nearly the same for a rename in the same directory as it would be for moving to a different directory (but rename would only need to write one directory entry rather than two.)
I do not know how movefile() or copyfile() are implemented in MS Windows. It is not clear which OS you are using; the term "volume" is used for both MS Windows and Mac OS-X.
