I'm more concerned with what earthly forces could have possibly broken that hinge. There just isn't that much stress on them normally. Doors are lifted from the bottom by the edge cables, so when the door is down, the door panels are merely stacked and the hinges just help keep them there, resiting only wind loading. When up, they support half the weight of the door panel (maybe 10 pounds). I suspect your door has serious problems and is tearing itself apart because of those problems.
"But that can't be! Twice a day I lift that door by hand. The action is smooth as silk and I can do it with one hand. If the door was binding, surely I would have noticed.”
Oh wait. You have a power opener! This has made you blind to problems that have been developing over the past few months, binding worse and worse, overstressing other parts and damaging them. The problems are now severe enough to kill somebody. Doors are dangerous. And openers make them a lot more dangerous exactly this way.
If it were me... First thing I would do is get a set of winding bars that fit, and unload the spring, i.e. Slack the lifting cables but not enough for them to jump their pulleys. Mind you, I know how to use winding bars and have a healthy fear of doors. I don't think there's anything wrong with your spring -- I just want to unload spring tension to make it safer to change hinges.
Next I would change hinges, one at a time, worst first. Because you still need the hinges to hold the door together. This is best done on a calm day, the door panels are stacked by gravity, the only load on the hinge is wind load.
They make tougher hinges. However the door would still have failed, just somewhere else that's harder to repair. These cheap hinges kinda did you a favor by failing first, since they are easy to replace. Each hinge has an offset, postioning the roller farther from the door the higher they go. Look for a number on it, e.g. 1, 2, 3, etc. The numbers are standard "hinge numbers" which reflect the offset, not the positions on your door.
With me so far? Now the real work begins.
You have fixed the symptom. Now fix the door.
Reload the spring (don't lift the door with the spring unloaded; even if you were able, you would just slack and birdcage the lifting cables, and then the door would drop.) And disconnect that darn opener.
Now lift the door by hand, cautiously. It is better to lift it from outside because the frame of the building will keep garage door panels from falling on your head. Just for reference, it should go quite smoothly. I have 12x14 industrial doors I can open with one hand.
My guess is it won't go quite smoothly, and you will find your door has a serious problem, or a variety of problems that are stacking.
Leave the setscrew wrench in the toolbox. Every time I messed with pulley timing or spring tension, I regretted it. If a pulley really has slipped, there will be subtle forensic clues of movement on the shaft, or the two pulleys will be visibly out of phase with each other.
If the door is trammed, look for lifting cables (on the ends of doors) that are frayed or broken. A stretched one will give near as many problems as a broken one; both will lift the door unequally, tramming it and causing a lot of binding. Replace both cables with a new matched set.
Also look for any rollers that are seriously messed up, but don't worry so much about them - a merely wobbly or dirty roller won't bind a door. If a pulley isn't entirely broken, I would replace it after the door is otherwise fixed.
Your spring is not broken, so don't waste your time thinking about that. The line of spray paint affirms that it's working. People always run to "it's a spring problem" and it rarely is.