I bought my Elite shortly after its release date, but it's not specifically an Xbox 360 issue since the PS3 does the same exact thing. It's crops the 1080p image with bars on all four sides instead of displaying properly at 1:1. Only the DVI port behaves properly.
So you got the old Elite when it was still a limited edition. If the rumor is correct, then new Elite with Falcon chipset should solve the HDMI problem, that is if the Elite is also replaced with Falcon chipset like other new models (Pro, Premium and Halo 3 Edition all have HDMI output now).
My Philip DVP5982 DVD player also has problem with displaying 1920x1080p properly using HDMI while it is fine using DVI, so yes it is not only limited to PS3 or XBOX360. My nVidia 7800GT video card only automatically detects FP241WZ's HDMI to be 1920x1080p (displayed properly) and not 1920x1200p. Therefore we can fairly assume that the signal processing chip for HDMI is different than DVI, but for some reason it can only properly process PC's HDMI signal and not video HDMI signal. I am wondering if there is something to do with different updates of HDMI (1.1, 1.2 and 1.3). If someone can figure out which chip is used for HDMI video processing that would be a great help. There is a reviewer on Chinese website who actually disassembled a FP241WZ all the way to the inside panel has nice pictures and descriptions of each primary chips. Just can't remember the name of website....