Have you checked the frame rate to see if the camera did create 25fps video, or if the video frame rate was of a more variable nature?
I ask, because I'm fairly sure my Samsung smart phone adjusts the frame rate according to it's surroundings. I have a Plasma TV, and if I point the camera at the TV while it's refreshing at 60Hz, I can see the rolling shutter/strobing effect at first, but the camera quickly syncs itself to the TV and it goes away. When the TV's refreshing at 50Hz the same thing happens, and I think the frame rate changes to 25 or 50fps.
I don't have any video handy that was shot near my TV while it refreshed at 50Hz, so that's just what I recall happened years ago when I tested it. Maybe it has something to do with the strobing vanishing from your tests. How many times does the projector flash the projection light for each frame? Is three times a standard thing?
Does anyone know how cameras go about syncing themselves to a flashing light source, assuming I'm remembering correctly and that's what they do? They'd have to do the same when used under florescent lights., wouldn't they? Florescent lights flash in time with the power cycle, don't they? I can't remember.