How to remove T spline

I'm not sure why you're having a problem removing the spline from your screen frame. I personally have always used a small screw driver, like you used, or a type of awl like the 75-029 and a set of small long nose pliers like the 75-055. 

There really isn't much of a trick to it, though. I would check to see if your door is in fact a Larson storm door. You could have the Forever door brand storm door which takes the 55-109 rigid screen spline. That's a little trickier to deal with. I've also related some videos that could help you determine what you need to do.