The handle has a "C" spring with ears, on the underside. In fact, window cranks AND door handles have the same setup.
You should simply slide the tool under the back, and catch the ends of the "C". When you do, keep pushing straight in. You are trying to spread and push the clip out the opposite side of the handle with the tool. Twisting will only get you in trouble, and it will damage the spring, (making removal MUCH harder). There is a learning curve. Sometimes you will get lucky. After doing a few, you will visualize what's happening behind the handle, and they get easier. Remember, the tool works by pushing on both ends of the clip, at the same time. It spreads the clip, and pushes it out of its groove. When that happens, the handle should pop off the spline. Save all your parts, and don't damage the door panel.
You can get extra springs everywhere, cheap.