It doesn't matter what kind of saddle you have, you can fall off. IMO your instructor should be looking at your seat rather than getting a "more secure saddle". If you want something to hold on to, get a neck strap.
Your friend moving barrels shouldn't be the end of the world - heck even our racehorses have to deal with dogs running around, kids, ponies, and ride around cross country and show jumps. Fair enough the horse got an initial fright, but I'm thinking it is you who is anxious about the moving barrels, not the horse.
Besides all of this - if you want to get a western saddle, get one. Don't pay any notice to what your friends are saying - do what is right for you because it isn't them who has to ride the horse.
Stop for a minute, open your mind, learn. You may not agree with what I say, I may not agree with what you say but we will both learn something new.