As I recall there is an option to change the smoothing behavior that releases the corner boundary verts from being pinned. You may have turned that on.

Or maybe the default behavior has changed between versions of maya, although I doubt this as I have been using maya since version 1.x and I always remember smooth working as I showed in the video. What version of Maya are you using? I am pretty sure that you have changed the default behavior though.

You might try renaming (do not delete) your maya preferences folder and then reopening maya and see if smoothing a square poly plane rounds. If it does then I have no idea why it is working that way on your system perfecto. I am pretty confident that is not the default behavior though.