Pretty sure this is either an issue with the Xbox as Rabbit73 suggests ..or with the television, itself. Given the age of the display and the fact it works with the disc player but not the XBox, there's also a chance it's in the HDMI implementation. Early HDMI rollouts were as flaky as the early USB rollouts. Supposed to be universal but some things worked and some didn't. Microsoft likely didn't test the XBox with displays this old.
Check the XBox on another display using HDMI. If it works on another display but not on the CRT, then it could be an HDMI handshaking issue. The XBox doesn't like what it sees during handshaking and shuts off the HDMI output. If that's the case, you might have to use the component inputs, instead. I see XBox 360 component cables on Amazon
and Walmart, but I don't own an XBox so I can't vouch for them. If it works, it's a pretty inexpensive solution.
It's possible a firmware update could also solve your issue. Trouble is, those firmware updates tend to disappear once a device reaches a certain age. I have a firmware update and a special cable for a Samsung SIR T-151 tuner I probably loaned out a dozen times in the last decade because Samsung ended support (and thanks to all the honest AVSers who mailed it back once they were done). I also have three Sony LCD displays I bought for next-to-nothing because they "didn't work." Firmware updates fixed 'em all.
Check your CRT's firmware version if there's a menu that'll display it. Then dig around on the web to see if there's an update. Probably won't be on Samsung's site, anymore, but someone somewhere might have it. Today's smart TVs can do that all by themselves. Sets this old will require a program on a USB stick stuck into that service port.
In the meantime, I'm going to retitle and move this to CRT displays where there's a better chance of someone with this line of displays will see it.