The twist tie in the photos (post #61) is only to show the final position and shape of the spring within the handle. It is not needed for assembly. Just push the spring into the handle as far as it will go. It will automatically compress and assume that shape and position.
You can't push the spring too deep. It hits bottom. You may need a screw driver to push it all the way in. Neither pin (725 52 93-56 or 501 87 49-01) goes through the loops of the spring, as it may appear in the parts diagram. Both pins are in front of the spring, as shown by the photos. Make sure the spring is fully seated, past the first pin hole. Just jam it in as far as possible. Push. The spring will stay in the handle by itself.
The tip of the "binder-actuation link" 501 87 53-01 touches the end of the handle spring (2nd photo). When the brake trips, the tip rotates counterclockwise and compresses the handle spring further. When you pull the handle to release the brake, the handle spring helps push the tip back into the off position. If the handle spring is not in place, you may have trouble resetting the brake.
BTW, you can easily test the brake. With the saw off, hold the saw level and about 14" above a solid surface. Let go of the front handle and let the tip of the blade free fall. Allow the saw to rotate freely about your trigger hand (limp wrist). When the tip of the blade hits, the brake should engage. If not, you have a safety problem.