Why can a wide-angle camera show pincushion distortion in air but barrel distortion underwater?

What is causing the camera to have pincushion distortion in air and then barrel distortion in water?

In your own question, your speculation is exactly correct: "Currently I am thinking that a change in the medium and thus a change in refractive index interacts with the wide angle lens and flat pane in front of the lens to change the distortion."

It's simply that the refractive index of water is different of that from air. The wide-angle lens doesn't "interact" with the flat plane in front of the lens, so much as a wide-angle lens will magnify or make barrel or pincushion distortions more apparent. Essentially, the water in front of the lens acts as another lens element, with its own refractive index. And adding lens elements to a lens changes its overall optical formula.

I would love to hear any thoughts on this and/or would appreciate getting pointed in the right direction of some research that may have looked into this already.

As far as "research that may have looked into this already", well, that's pretty much the entire history of optical physics. This is very simply a result of immersing the subject in a different optical medium than what the camera lens + imaging system exists in. That is, the water surrounding and in between the subject and the front lens (or lens protector / waterproof case or element) results in a different angular magnification in the scene than you get when everything is in air.