If the only time you ride are during your lessons, my suggestion won't help you since they are all riding homework things.
First, to answer your question about whether you should be gripping with your legs or standing in the stirrups, the answer is no. By that I mean neither is right. Ultimately what you need is positive friction. It's easiest to learn by riding without stirrups. Unfortunately it's a feel more than anything, so it's very hard to explain in words. It's a happy medium between sitting there with your upper body collapsed and your legs flopping around like noodles and sitting with a rigid body and legs clamped on. It's a combination of balance and using your inner core muscles to stabilize your own body to not overreact to the horse's movement.
Now that I'm completely confused you, my suggested exercise for strengthening your position and finding your proper balance point for 2-point is, vary the amount of time your stay in the up phase when posting the trot. At first post normal. Then stay up 2 beats and down 1. When that gets easy, stay up 3 beats, down 1. When that gets easy, start mixing it up. Even sit a few beats. What this does more than anything is help you find the proper balance point over your feet. So you don't catch your horse in the mouth, grab mane when you first start doing it. You will most likely fall forward/backward the first few times until you find the sweet spot. Once you've done the exercise enough times to where you think you've got it, keep your hands on the mane as a backup, but let go of any real tension on it so you are effectively holding yourself up. Hope that helps.
You just have to see your distance...you don't have to like it.