"And does NOT make the Movie Maker Project file independent or self-contained."
That's quite normal, in video editing programs, but this from Microsoft:
"A project holds information about the arrangement and timing of the items that appear on the storyboard/timeline, which include audio and video clips, video transitions, video effects, and titles. It does not contain the actual audio, video, and picture source files, but rather references them as clips."
Also: "You can open Windows Movie Maker project files that have been created and saved in either previous versions or in the current version of Windows Movie Maker. However, after you open and then save the project file in the current version of Windows Movie Maker, the project cannot be opened in previous versions of Windows Movie Maker."
Thus the project file should be OK for you on both desktop and laptop, as you now have the same 2012 version on both computers, but only if the files are identical and the paths to the files in the project file are identical and the name of the destination folder must be the same, otherwise the project file is looking in the wrong place. At the moment, the destination is different as between your desktop and laptop. Hope that all makes sense.
I only have WLMM 2011, I am assuming the versions work just about the same.
The project file for me is saved with a .wimp extension. It can be opened in notepad and will display the filepaths, so you can see what I am getting at.