Personally, I would use one version of a background image, making it high enough resolution to look nice on all screens. Your file is currently 1200px wide but still only 59KB, which isn’t bad.
But I would also split the image into two: the rain in one, and the woman in the other. The rain doesn’t have to be as high resolution as the woman, and you’ll have an easier time placing her in the right position when she’s on her own.
Not to mention that half the population do not maximise their browser so those sizes are a bit pointless. I never have my browser maximised and often have two or three browsers open at the same time.
The approach you need to take for responsive design is to create images that are fluid enough within a certain range so that you don’t need hundreds of them. In your example I don’t see a need for more than one image unless you want the higher resolution devices to get better quality or oerhaps a smaller image for art direction on smaller screens.