All the videos of painting elephants I've seen include a handler that very clearly is holding their tusk or ear. I'd absolutely love this to be true but it quite honestly seems like puppeteering?
The handler is obviously present to manage the brushes, I presume the physical contact is more to do with signaling when to break for reloading/switching brushes.
Do you really think the linked BBC clip would have been made if it were "puppeteering"? The whole premise was working with the elephant to explore its creative side.
There are possibilities beyond intentionally controlling the elephant. There is for example the case of Clever Hans[1], a horse who was supposedly able to perform arithmetic: it was eventually concluded most likely that Hans was reacting to his trainer's unconscious cues.
Do you really think the linked BBC clip would have been made if it were "puppeteering"? The whole premise was working with the elephant to explore its creative side.