A type of compartment that rises out of a desk
Search for &quot;Lift up&quot; hinges. These will let you lift up a section of desk to reveal something underneath, but won't let you click in the desk to lift it up. For that, you'll want push latches.
I'm not quite sure how well those will work in combination. As a fallback, consider some kind of recessed handle. (There's a bunch of options out there.)