There is no automatic renaming or moving files in the version control systems. Working with the clients, you have to manually copy a file to a new location, change its name, add to VCS and remove the old file. WebStorm's refactoring features make it possible to easily move and rename files under version control, performing all actions involved in these functions.
In the Version Control tool window, next to the resulting files WebStorm explicitly states: renamed from <file name> or: copied from <folder name>.
To copy, move or rename a file under version control
Select the desired file in the Project tool window, and perform refactoring procedures, as described in the sections: