You are sending the commands over the network but the files are moved/copied directly on the NAS. If you use two Windows Explorer windows to move files from one SMB share to another one, the file data will travel to your desktop system and back.
I believe cross-share copies should occur server-side (assuming client properly supports server-side copies). Perf is even better in case of block cloning.
Samba preferentially uses copy_file_range() which is indeed limited to same filesystem, but if that fails it will fall back to opening both files and pread() from one and pwrite() to the other (e.g. implemented in userspace).