Sauté the cabbage and onion in the oil in a large skillet over medium heat, stirring them often, for about 5 minutes. Reduce the heat to medium-low, and sauté the vegetables 5 to 10 minutes longer or until the cabbage begins to turn golden.
Add the carrots and garlic, and sauté the vegetables for another 5 minutes. Remove the vegetables from the heat, and let them cool to room temperature.
In a large bowl, combine the beef with the egg white and whole egg, bread crumbs, basil, oregano, salt, pepper, water, and vinegar. Add to the beef mixture the cooked vegetables and the shredded potatoes, and mix the ingredients well. In a shallow baking pan, shape the meat-and-vegetable mixture into a loaf about 8 inches long. Cover the pan tightly with aluminum foil.
Bake the meat loaf in a preheated 350° oven for 30 minutes.
Uncover the meat loaf, and bake it 30 minutes longer.
Uncover the meat loaf, and bake it 30 minutes longer.
Let the meat loaf rest for about 15 minutes before slicing it.