OBS Studio is a full rewrite of OBS, and the old version was renamed to Classic to avoid confusion. It's been deprecated for many years now, and is no longer maintained. There is no real reason to use it anymore. If I had to guess the issue here, you're on a laptop with two GPUs and OBS Studio is running on a different GPU than OBS Classic is, which is why you're running in to issues. We'd need a log file to know for sure, but it's likely a simple fix (which you can stop by our Discord or forums with and we can take a closer look, feel free to ping me @fenrir in either location).

If you'd rather do your own testing, here's a starting point for my guess as to the issue: https://obsproject.com/wiki/Laptop-GPU-Selection-Windows-10

Desktop system, only one card (GTX 1070). Thank you for the starting point.

If you can stream your games on a Zoom/Skype call as a test, then OBS should be able to handle it too with the right configuration.

On those pesty games, neither Zoom nor Skype, nor TeamViewer capture the screen. I have also my own, written long time ago, screen capture utility, that one fails too. PrtScr key fails too. All of them return the correct resolution in cliboard, full black. That's why my guess is a DRM problem.