Your second option, with slight modification, is what's often called interval speed work. From the Runner's World article "Speed work for dummies":
Intervals: These can consist of longer (800m-2000m)[0.5 to 1.2m] or shorter (200m-400m) intervals. Intervals are run at about 85-95% of maximum heart rate. Longer intervals: run at about 5km race pace, with recovery time equal to or slightly less than the interval running time. Shorter intervals: run slightly faster than 5km race pace, with longer recovery periods, 2-4 times as long as the repetition time.
The recover time is the main difference from what you describe -- keep it a bit shorter than the interval time, and don't rest completely -- jog slowly or walk. If you don't know your "5k pace" (the maximum speed you could maintain without stopping for 5km / 3.1mi) just run hard on your first interval, then adjust faster or slower depending on how it goes.
There are numerous advantages to this type of workout. From another Runner's World article, "10 Health Benefits of Speed Training That Go Beyond Faster Times":
- A stronger stride
- More muscle
- Better fat burn
- Reduced risk of injury
- Stronger bones and other connective tissues
- Improved running economy
- More anaerobic endurance
- Better balance and proprioception
- Improved agility
- Easier strides
Here's a few key points under better fat burn:
At what pace do you get 100 percent of your energy from fat? It’s a trick question because the answer is this: You burn 100 percent fat while standing still after a sprint, resistance exercise, or other high-intensity effort. When you jog, only about 70 percent of your energy comes from fat. During a distance run, it’s about 50 percent. At mile pace and faster, you use 100 percent carbs for energy. But after those high-intensity efforts, your body spares carbs by using 100 percent fat to resupply your first-response anaerobic system and to fuel recovery.
So during an interval you'll burn more calories, then afterward you'll burn more fat. In addition, intervals increase your body's muscle mass, which increases the amount of calories you'll burn while doing no physical activity. So in that sense, interval work is a win/win.