I see thanks.
So if i understand you correctly, you want to make sure before buying the recorder that both video and sound will record. But by selecting DVI in Xbox HDMI audio is disabled, if you select HDMI video isn't passed on to the monitor.
DVI and HDMI are both digital and is the same signal. Its how the XBox is communicating with your monitor that is missing when set to HDMI because of the adapter cable. Try forcing the resolution if you can by not having it on auto detect.
http/support.xbox.com/en-US/xbox-one/console/display-troubleshooter
To use the display troubleshooter, follow these steps:
Check the current resolution setting on your TV.
Set the resolution to 1080p if your TV supports this resolution. If your TV only supports 720p or 1080i, set your TV to 720p. Then, on your console, select the matching resolution on the Change your display screen.
If your resolution is set to Auto Detect, select 1080p from the Change your display screen on your console.
There is a possibility the elgato accepts Xboxs EDID communication (like talking to a tv/monitor without a display) and the elgato then just spits out (pass through) the video feed out to the monitor regardless if using dvi adapter, like tricking the Xbox. But not sure on this!
Its a tricky dilemma, maybe get a HDMI monitor with speakers?