How do I determine whether I need a standard toilet wax ring versus the thicker wax ring? Is there a gap distance that would suggest one over the other. I installed tile and now have a 1/2 inch gap between the flange surface and the floor (tile) surface.

So the top surface of your flange is 1/2" below the top surface of the tile, right? And I am assuming this is not a moveable flange like a Twist-N-Set. You are in the market for an oversized wax ring or you can just add a PVC flange spacer. The falnge spacer is little more than a PVC washer that matches the holes in your old flange, I'd go with the spacer and a regular wax ring.