Has it ever been clarified what TES means by "soul"?
No real consensus can be reached, as the effects of soultrap and enchanting hasn't been consistently depicted.
True, but ignoring Oblivions mechanics and focusing on the lore from the other games there is a constant suggestion that a soul gem captures the soul of a being rather then it's energy.
A soul can be defined as some sort of connection between something nebulous and the body. When the body is destroyed, there is nothing to keep the soul in place and it returns to where it came from. For a Daedra it means they are remade in Oblivion, a mortal is reborn in Mundus.
So when a soul is trapped, it can be said that the body is replaced with a Soulgem. When the soul is then used for enchantments the item becomes it's new body( as in Feyfolken). Necromancy would pull back parts of the soul that belonged to the bones and force them to animate their old bodies (hence intelligent creatures make better servants).
The nebulous object attached to the other end of the chain can be described as the alligned-AE from the loveleter. Albides didn't clarify but this can be something like a Daedric Prince, the Dreamsleeve or Paradise.
It is possible to change your alligned AE. Battlespire had a few game-over choices where you could join the Daedric side and become a Daedra and Mankars followers were reborn in Paradise. When becoming a god, you become your own AE, Vivec had his provisional house, Mankar had Paradise. I also assume the Daedric princes have their own AE.
It's also possible to follow a dying soul to his afterlife. For example in the Trial of Vivec he hitches a ride the soul of a Khajiit (try to see the image!) into Moonshadow to hit Azura where it hurts. So perhaps one of the alternatives to entering Paradise would have been to capture a follower of Mankar Camoran and ritually execute him.