If played by anything other than your foot, the goalie can pick it up.

The better question is if you change thigh to foot what happens in this situation?

"The other team hits a corner kick into the box, where a defender plays it off of their foot into the air."

Then you go to the 2nd part of the rule, is it deemed deliberate, rather than a deflection or miskick?

On a corner kick, I would think what you are describing would most of the time be considered a deflection off of your foot, especially if you hit the ball with your foot while in the air. Kind of depends on if anyone is around you or not and if the ball is bouncing or rolling slow...definitely up for interpretation by the ref.