Hi Mumbles -- yes, you really need to be relaxed in your hips and your hip flexors. You need to sit on the "pockets of your pants" and keeping the pockets of your pants under you, you need to let you hips open and close from your hip flexors. It takes relaxation. But it also takes cooperative hip flexors!
Now, of course, that is much easier said than done!
When you're learning and the instructor has you in 2-point, you have your hip flexors counter to where you'll need them to be. When you're in 2-point, your hip flexors are closed (you're leaning forward bent from your hip flexors. In 2-point, your hip flexors stay in that closed position). When you're sitting on the pockets of your pants, you need to let the motion of your horse take your hip flexors through a reasonably full range of motion. They will close as she/he takes off, and then they will open to a peak point as the horse rolls out under you. (The horse is moving in a wave-like motion, and you need to move in that same wave-like motion.). Keeping your pants pockets under you, you need to allow your hip flexors to open and close with the wave motion.
Learning in 2-point is not a bad idea, as it gets you accustomed to the motion of the horse and the rhythm of the canter. However, you then need to sit down and let your hips go. That can be really difficult, because a lot of us have closed down hip flexors (especially if you work in front of a computer all day). So know that some of this is just physical -- if your hip flexors are tight, you have a true physical barrier to being able to move smoothly with the horse.
The school horse that you're riding sounds really lovely, and I do believe that she is working very hard to keep you balanced. She feels that you're not in balance with her motion, so she stops cantering. (I say this, because I know it first hand. I had this very same problem for a very long time. I even got to the point at which I was convinced that I just could not do it. I got psyched out!)
The only advice that I can offer is to stop 2-point (since you likely already are at the point at which you understand the rhythm of the gait) and then to try as hard as you can to relax your hip flexors while she canters under you. The more you can relax those hip flexors, the greater number of strides that she will be able to get before she stops. Each time you can get your hip flexors to move through the full range of motion is a time that your hips will start to loosen up and learn the movement (muscle memory). Over time, you'll start getting more and more strides. Then one day you'll suddenly realize that she's no longer stopping, and you'll be cantering!
Be very easy on yourself with this (e.g., don't get psyched out!). If you start to be mentally hard on yourself, your hip flexors are going to seize right up! (I still suffer from "the canter psych". If I ask for a canter and it doesn't got well, I instantaneously go into mental failure mode -- "I am not capable of cantering!" I get rigid and stuck. However, then I roll my eyes, laugh at myself, and relax. At that point, we can canter just fine.) So I warn you in advance not to do this to yourself. Be gentle with yourself now, and it will serve you well for all of your canters to come!