After watching several videos of elephants painting, it's obvious that the elephants are doing the painting themselves, but it appears that they are trained to paint specific pictures. I saw some videos in which a trainer would put the brush in the elephant's trunk, and then, holding the elephant's tusk while it held its trunk steady, he or she would guide the elephant's head to make a particular drawing. I also saw some elephants make the drawings without any assistance from a trainer, so I'm guessing that once the elephant is trained, it is able to produce the paintings without any assistance.

So I guess what this means is that in the case of some drawings at least, elephants don't have all that much artistic ability, but I think that if they can be trained to paint realistic images (and many people also need to be trained to produce realistic images), this nevertheless indicates a fairly high level of intelligence.