A saddle that actually fits you and your horse is going to help you the most.
Other than that, your hands are actually very correct in their positioning. A rider who is learning (I include myself in this category!!) should have their knuckles resting on the crest of the neck, near the withers. You do however need to keep the reins a bit shorter and keep your hands closed around the reins. Your chest needs to proudly push forward and your shoulder blades should slide down your back into your pockets. Once this happens your body will naturally root down into the saddle and your hands will become more "forward thinking". Then you need to sit and drive.
At this point you do not need to worry about where the horse's head is and simply focus on balancing your own position in the tack (hopefully in a new saddle!!).
@anebel: thanks for your comment, I will try to work on everything. But I don't think I'm going to buy a new saddle soon because she's not my own horse. I wish she were, then I'd take all responsibility. I do hope the riding school will provide a new saddle shortly.
@DressageDreamer: I hope I can let my friend take some more pictures this summer so you can see if/how I progress:) .