I can assure you that modified DVD firmware is not the cause of your corruption.
I would guess that the box that reads the saves just fine is on a much older kernel. There used to be plenty of achievement/gamesave hacking going on earlier in the 360's life, but after a certain point, MS said it was allowing people to cheat as well (take DOA4 for example), so they updated the kernel to sign gamesaves and require a valid signature on them. Your modded (and offline, so not privy to immediate updates) console, if it's on an older kernel, would be saving the required information, but not signing it, so when your elite (which is probably running the latest kernel) views the data as "unsigned", it returns as "corrupt" (you can verify this by breaking the signature on any hdd content, and subsequently attempting to load it). You can try updating the modded console (you don't have to use the newest kernel; just whichever one they started the gamesave thing, which I think was pre-4552) to see if this eliminates the problem.
Also, the reason an elite won't show 4 red lights when the video cable is unplugged, is because it has an HDMI port, and tmk HDMI ports don't use a monitor ID (just like composite/component/etc). On a normal 360, if there is no cable (ie, no jumpers set), it assumes you cannot have it connected. This makes me wonder if it's possible to run a component signal to one TV, and an HDMI to the other... I might need to test that for fun.