Mfair5,
In my experience, all of my horses "try" to accellerate to a canter or lope when going up hill. It is easier for them to get power behind them to climb the hill. Your friend probably did not know how to slow your older horse in such a situation. It was a natural reaction for the horse you were on to want to follow. Being herd animals, horses never want to be left behind their fellow horses.
I had a similar situation many years ago before we owned our horses. I was trail riding with an acquaintance who suddenly took off at a gallop, never thinking of the situation I was in. I held back the horse I was riding, to wait for another rider. The horse became a bucking maniac! He didn't buck me off but the incident caused me a back injury which I nurse to this day! I wish I had known back then how to do an emergency dismount. 
I now always teach less experienced companion riders a mimimum of the one-rein stop, how to stop a horse from performing unwanted actions by circling, and how to give the horse cues he is trained for. Whatever the issue is with a specific horse, I try to prepare my fellow rider in advance.
The other posters have given you some good advice. You did a great job! There usually is no substitute for experience. That's how we learn. You don't have to be on a steep trail to work with your 6 year old in holding back when a companion horse takes off. I would do that.
Good luck!

No heaven can heaven be, if a horse isn't there to welcome me.