There are a few ways you can do it. Basically, you need to get your photo into a format that's suitable for using as an environment map. There are a number of such projections available. The most common ones I've run into for 360° video are:
- Cube maps - 6 images that show what the camera would see if it were looking forward, backward, left, right, up and down from the same position.
- Equirectangular - the 2nd image you show above is this type of photo. It must be exactly twice as wide as it is tall. The rows and columns of pixels then correspond to the latitude and longitude of the sphere you map it onto.
- Dual hemispherical - this is when you take a camera (real or virtual) and make an image with a 180° field of view. There are 2 - one for the front and one for the back.
Given the image you have, there are a few ways you could create something useful from it. You could crop it or extend it so that it has a 2x1 aspect ratio and use it as-is. You'll get a weird seam where the left and right meet, but if you are handy with photo editing tools, you can probably modify it to look reasonable. Note that this will give you the correct image for an equirect projection, but it may or may not look how you want. That purple band would never be visible all at once. It would wrap around the entire field of view, for example.
Another option is to make a cube map out of it. You could crop what you have into 3 equal squares for, say, front, left, and right. Then you'd need to generate the other 3 yourself. Generating a star field isn't too difficult either in code or using a photo-editing/compositing tool. Or if you know any artists, perhaps they could help you.