I recently ordered a new MacBook Air and an adapter for an HDMI input. Now I wanted to know if I could connect my Xbox one x to the Mac.
No, the MacBook does not have an HDMI input even with the adapter. This is an adapter for an HDMI output.
Yes, you can connect the MacBook to the HDMI input of the Xbox, then the Xbox will show your MacBook screen.
The other way around is not possible ^^