Thanks for the info.
Yes you are pivoting on your rocker (the slightly curvier part of the blade). With a 2-foot turn you are doing a forward outside (FO) 3 turn with the inside foot and a forward inside (FI) 3 turn with the outside foot. Going into the turn your inside foot should be on an outside edge and your outside foot on an inside edge. The amount of lean on each should be pretty much identical. What most beginners have trouble with is getting the inside foot on an outside edge. Try really focusing on the edges you are on when doing a curve. With a two foot turn you do 2 pivots, simultaneously, one on each rocker. I'd have to try one to be sure but I think the feet should be parallel so neither foot too far in front or behind. By lifting your inside leg after the turn you are simulating the exit from an FO 3 turn. If you were to lift the outside leg after the turn you should be on an outside edge with the inside foot and you would be simulating the exit from an FI 3 turn. (On exit from the 2 foot turn you should be on an outside edge with the inside foot and an inside edge with the outside foot, so opposite to entry).
What happens when you turn is that your free side (the half of your body on the outside of the circle) comes round (forward) which pushes your body to rotate around the blade or blades. The reason you are going on a too-tight circle if you take more speed into it is that you're not used to stopping the rotation of the body so your free side keeps pushing you round. You will need to turn to "check" the momentum of the turn by using your arms, shoulders, core muscles, lats, obliques, to stop yourself rotating round out of control. Try searching on YT for checking a 3 turn. It's hard to explain (for me anyway). Sinking down into your knees will help you. Eventually stretching the free leg back rather than letting it pull you round, but I suspect at the moment your coach wants you to keep it near the skating leg. Inside edges are hard to check, even for experienced skaters.
I think the general aim of 2-foot turns is to get you used to turning without the fear of skating on one foot. Lifting the leg after the turn is a preparation for doing the one foot version. An alternative is to go into the turn one footed and come out two footed, which can help if you're not good on back edges on one foot.
Even a 2-foot turn is hard to do well and there are many factors at work. Try not to be discouraged and enjoy the process. I've probably given you too much information.