In a 1-gallon zip-lock plastic bag or a large bowl, mix vinegar, ketchup, soy sauce, oil, Worcestershire, honey, mustard, garlic, dill, and pepper. Add steak, seal bag or cover bowl, and chill, turning meat occasionally, at least 4 hours or up to 1 day.
Lift steak from marinade and let drain briefly; reserve marinade.
Place meat on a grill over a solid bed of medium-hot coals or medium-high heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds); close lid on gas grill. Cook steak, turning to brown evenly, to medium-rare (cut to test), 10 to 12 minutes.
Transfer steak to a platter and keep warm. Pour reserved marinade into a 6- to 8-inch frying pan and bring to a boil over high heat. Pour into a bowl. Cut meat across the grain into thin, slanting slices. Serve with marinade to add to taste.