As I understand it, soul and spirit are not the same thing. The soul resides in the blood, and “sleeps” with the body at death. The soul is the specific memories, behaviors and personality traits that are unique to a given person. The spirit goes to the presence of God at death.
At resurrection, we are judged body and soul, as the two are inseperable. Those unworthy of eternal life in the eyes of God experience the second death. Afterwards, death itself is thrown into the lake of fire and annihilated.
The popular conception of Hell is a mishmash of Sheol, Gehenna and Tartarus/Tartaroo. The Bible states that the flames of eternal torment are for those rebellious angels and evil spirits that were bound in the deepest depths of Sheol.
I’m not going to go out on a limb and say that there is no punishment other than annihilation after death for sinners, however.
The spirit, as I understand it, is the life force of the person, which came from God and returns to him at death. It is not the individual personality or memories.
Possibly this doesn’t differ that much from your definition.
I wouldn’t say the soul or individual sleeps at death, rather that it goes out of existence. It can, of course, be brought back by God, if he so chooses.
It's Apocryphal, but I remember some passages from the book of Enoch that cover the 7 pleasures and 7 torments...the torments being those that didn't make the cut are separated out after death and exist in a state removed from God, yet, can see Heaven, can see people there living blissfully, and know that they themselves will never experience that joy. I can't imagine a more tortuous state of existence..it is apocryphal, but it makes a ton of sense.