@pachyderm pro I don't know that software. Can it do automatic edge detection? Can you use a harder brush for burning in? Those features would make it easier to avoid darkening the parts of your image that should stay bright.
For a picture such as that elephant, I would recommend keeping the background because it would greatly complement the action. The only times I darken a background is when the background is distracting, bland, or already dark, because in most cases the environment the animal is in separates a great image from an exceptional one. I would also recommend brightening the elephant a bit so that it stands out and is clearer to the viewer. Editing usually isn't easy, but with great practice you will have some great shots. Until you become very familiar with your software, I would advise against attempting to blacken the background of an image, because it is very difficult, especially in an image like this one with a lighter background.
I don't think either work, or look particularly attractive! They could both easily be improved by not removing the background... If not, by making the elephant more visible - removing some of the brightness of the sky and the grass, would make it more noticeable as these are the things which immediately stick out.
