Though it’s mostly for show, I technically have an elephant trainer’s/ caretaker’s license from Thailand. One could go so far as to call me a Mahout.
The best I can say is that “training” elephants is a complex subject, but living and working with them is… holy.
There are many methods to training elephants and I chose my mentors based on theirs. Most elephants in Thailand are trained with elephant hooks/ goads, which I found most places abused the use of, and breaking in a wild elephant is something I didn’t personally see but by most accounts it’s barbaric. The best places, including the place I was taught, has a mother who lives with her family in the wild and they were all born within human contact so that breaking in is largely unnecessary (the mom naturally helps in taking care of her young; she makes sure they don’t stray, eat, harm humans [especially human children] accidentally, and she helps with other things, like scooping poop out of their bums). They just have to get used to you, it’s like working with horses.
Elephant foot chains are like dog neck leashes, you can use them to abuse or you can use them mostly to make sure that you don’t spend a whole morning searching for your lost one in the wild after a night of bad weather (waking up after a night of heavy rain is terrifying because an elephant playfully wandering around could have easily lost its footing on a muddy hill and fallen); the elephants I worked with still had a pretty big allowable field to play in, the motto was as long as someone could see them in the hills it was fine, and even when we couldn’t no one really worried (if an elephant’s mother isn’t panicking, you really have no reason to either).
I woke up and slept by them every day in their natural habitat, worked under the sun getting/ cutting their food, bathed and played with them in the mud and rivers, &c.. The ones I trained with had no seats on their backs so that helped from a humane POV, but there are other Mahout schools which I came into contact with daily that use the seats for tourism purposes and they have a lot of “caretakers” who use elephant goads just because they’re lazy – imagine teenagers working at an amusement park (I had to work at one of these as a guide for a bit, which felt horrible, but there’s nothing I could have done as a foreigner outside of refusing to use a goad).
In essence I don’t think you actually train elephants, you either break them in somewhat/ mostly cruelly, or you learn how to use your body, hands, and feet to move with them to the same ends (this is easier said than done, for example there are around 108 specific areas [marmans] on an elephant that need to be avoided during training; 44 on the limbs, 3 on the lower abdomen, 9 on the chest area, 15 on the back, 12 on the neck, 25 on the head, and other areas such as the anus, genital areas, mammary glands, trunk, heart, center of the face, eyes, ears, mouth and central pelvis area). I did the latter while I was working with these gentle giants, I honestly think they just saw me as “something that isn’t too bad to have around that occasionally rides us, picks bloodsucking bugs off of us, gives us food, and bananas as treats.”
I don’t know how to train or work with elephants circus-like because I didn’t make them entertain me, I just learnt how to move with them.