And this "server move" which WinSCP uses is something different than the normal mv command?
The common *nix
mvcommand moves the files, when they are on the same file system. This is the same operation that SFTP server "move" functionality does.
But then they are not, it does not (cannot) move the files. It copies the files and then deletes the original. Most SFTP servers do not support "copy" operation.
See https://winscp.net/eng/docs/protocols (section "Direct File duplication").