I've looked at pictures of me when I was a toddler and while I wasn't a scrawny kid, I wasn't fat. Somewhere along the line I started getting labelled the fat kid and teased. Each school photo shows me progressively heavier than the last. Somewhere along the line, I internalized the idea that I was a fat kid and lived up to it by, well, being fat. I've never confused the teasing for being fat as a cause. But it certainly didn't help me get into better shape. The teasing sucked, but it was tolerable. Instead, a couple of incidents that actually had me feeling worried for my physical well-being really sparked the change.
The bullying in high school hit a low point when I was doing a science presentation in a class the teacher could not even control. Two of the guys in the gang I referred to in my first post started tagging my freaking presentation board with gang symbols. It's a hilariously stupid scenario in hindsight. Honestly, I laugh about it now. But at the time I was freaked out, and angry because this meant I couldn't use it for the science fair later. It wasn't them teasing me about my weight that prompted me to lose weight, but rather fear of physical harm from them. I had to sit beside these guys in class and I'd see them come to school with knives and pull them out and pretend to stab people with them. In hindsight, they wouldn't have been dumb enough to hurt me so long as I didn't give them a reason. But not giving them a reason to hurt me felt like such a cowardly thing. It was letting them frame the "debate" and to hell if I would let them do that. It was never about me, it was about them and their issues with their father beating them or something.
So quite honestly, I was partly motivated to start losing weight out of survival instinct. I wanted to be able to defend myself if it ever got to that point, so I started pumping iron so I would be stronger than them should things ever get to that point.
The thing is, I wasn't even really the primary target of these bullies. So as ridiculous as my account above is, other guys had it worse. They really did get beaten up. There's a fine line between being pushed to the brink and being broken. I wasn't France in WWII, I was Britain. I had options available and I responded accordingly.
To your point however, my gym teacher also had a part in motivating me to lose weight. He would never cut me any slack. Jocks would get remedial attention in math courses they were having problems with. There is no special education for gym. In gym, when we had to say do gymnastics on hanging rings or flips on mats, I never got any slack. My teacher would make me get up there and fail, very publicly, and in a humiliating way. He gave me a C for effort.
Bullying issues aside, I think my school did exactly what it needed to help me out. The gym classes included lots of talks by our gym teacher about how we would all gain weight as we got older, got office jobs, and became immobile. And some humility is good. I couldn't even hang off of the gymnastic rings let alone do what was required to get an A. If this were an Algebra course, I wouldn't have been able to count. If anything, they were a bit indifferent about my situation. It's one thing to point out to a kid that he has problems. It's another to give him options about how to solve them. I stumbled upon weight lifting as an option only because my older brother was a wrestler and I'd seen him do it.
As far as the worst bullies go, that's a whole other kettle of fish. The solution to treating those sort of bullies these days is to shuffle them around the school system until they either play out the clock and graduate or drop out. My school was considered a "good school" which had the effect of encouraging principals at troubled schools of sending their worst offenders to our school to discipline them. So 99% of the students were good but had to face 1% of the school population which were probably psychopaths. I don't think schools can treat these bullies. They simply don't have the jurisdiction. For a lot of those kids, you'd have to remove them from broken homes.
For your casual bullying, I'd say the best approach is to play both sides of the coin. Tell the bullies to get over themselves and tell the victims to do the same. If you're that hung up over getting bullied for being 120 lbs at 6'0" tall, maybe it is you that has an issue with your weight. At that point, the bullies are merely amplifying your discontent. You can either learn to accept it and let the comments slide off of you because they no longer hurt, or you can do something about it and gain weight.
One of these days I need to learn how to be more concise :-)