Haha, during both videos I found myself clicking to get the horse to move :P You do appear to be blocking him from going forward, and that does not help make a lazy lesson horse move! I'm sorry to hear that your first canter encounter was unpleasant, but glad that you got right back to it!
While you do seem to have a more steady seat than many just starting out that I have seen, I agree that you are moving too quickly. When I started cantering it was on the calmest lesson horse (not the one I usually rode) on the lunge line and I did not have reins. I had been riding with a cruddy instructor for about a year (one who basically ignored me in a huge group lesson), and probably another several months with my old trainer. Certainly not in a large open field! I would also be very wary that your instructor is willing to put you on an unpredictable horse, especially when going out of the arena. Are you paying for group or private lessons? If you are paying for private, then it does not seem fair that she could just say "hey, we're going on a group hack!" unless you were on board. I don't know what other options are available around you, but you may want to look into taking lessons with a different instructor if possible. In this sport you want to be training with someone who will set you up for success- your trainer may be a lovely rider, but not every great rider is also a great teacher!!