If the default game capture mode is unable to record a game (in other words, you don't see the overlay appear), try fullscreen capture mode instead.
Fullscreen capture mode will simply record whatever you see on the screen and is ideal for capturing older games that don't use modern versions of DirectX, webgames from browsers and when you want to capture from multiple apps at once (if you have overlays that you want to include, for example).
When you enable Fullscreen mode, the overlay will appear immediately and you can then start recording using the usual hotkey. (The exception to this is if you are running an OpenGL, in which case the overlay will not appear but you can still start recording)
Unlike Game capture mode, the overlay will then disappear while you are recording. If it didn't it would also be included in the recording which is particularly desirable.
Then, when you want to stop recording, either press the usual hotkey or click the Stop recording button that is available on the FBX window.