How to fix gap under front door? Threshold?

My front door has a 1/2" gap underneath it when closed, so it is quite drafty. I am not sure how to fix it. I put a door sweep on the bottom, but I ran into an issue. Forgive my lack of terminology, but I've tried to illustrate the problem in the photos. The green arrows show the area that the door sits over when closed. The red arrows show a "threshold" that is inside of that area. This threshold is about 1/2" higher than the green area. This limits how low I can put the sweep on the door. Right now, the sweep is as low as it can go and still close. However, you can see that when the door is closed, there is a gap underneath as the sweep isn't low enough to touch the green area. You can even clearly see the light coming through What are my options? Should I get one of those bean-filled strips and somehow attach it to the door? Or do I need to remove the red arrow part and replace it with something shorter? (This part seems to match the laminate flooring, so I am guessing a new shorter part wouldn't match, but oh well). I'd like to get this sealed up quickly as winter will be hitting before long in teh Northeast US. Thanks for any advice!