I figure I'll eat the bullet before someone else does. When you say "the correct way", I believe you mean "(a) better way(s)". Doing it not "that" or "those" ways is not wrong. There is no right. There are just utilitarian criteria according to goals and what way of coordinating your body best helps you personally satisfy them.
I would disagree with that concept of right/wrong. There is a right way and a wrong way. After 40 + years of the aikido world doing tori fune, it pretty much solidifies the notion that there is a correct way and an incorrect way to train. Otherwise, we'd have a whole lot of Ueshiba-level people in aikido. Now, just replace "aikido" with "taichi" and "Ueshiba" with a grandmaster level teacher in taichi and you pretty much have the same thing.
Ueshiba's torifune is ... interesting.
Starts around 3:40 and goes to around 4:02.
Notice how he changes from arms going upwards to another set where arms are more in front.
Notice Ueshiba's hands. Not just that they are in a fist, but look how they spiral from outward to when he has them at his sides.
Especially notice 3:54. Notice how his upper body is turned? Significantly.
Now, to compare:
Watch Terry Dobson. Notice his shoulders and his hands.
Or from this video:
Are these people moving the same way that Ueshiba is moving?
So, if aiki is all internal and no one explains what you're supposed to do, then how are you to do the exercises correctly? You can only copy the form and hope that you can figure something out.
Even if someone explains how to do this exercise, there is a correct way to train aiki and an incorrect way to train aiki.