So Joseph Smith and Elvis are both alive and well, huh?
Sort of. I think you are on the right track, but that the terminology is getting in the way of a completely accurate understanding. Although we occasionally use the words "spirit" and "soul" interchangeably, and then throw in the concept of "intelligences," that is not technically an accurate way of expressing our doctrine. I'll see if I can clarify it.
"Intelligence" is the "light of truth" that has always existed; we believe it is co-eternal with God. From this essence, i.e. this light, He formed the spirits of all things before creating them temporally. And when I say "all things," that is exactly what I mean. All of His creations have a spirit component of some sort, which is not to say that an animal's spirit is the same as a human spirit. We believe that each of us existed as an unembodied spirit prior to coming to earth and lived for billions of years in God's presence before being born. This may have been what the "Joseph Smith" poster was alluding to in his post.
When the spirit enters a physical body, the resulting entity is a "living soul." The "spirit" is essentially the life force that uniquely identifies each living soul.
The living soul continues to exist until death, when the spirit leaves the body. The spirit, however, continues to exist as a cognizant life force until the body is resurrected, at which time it re-enters the body, which is then made perfect and immortal. From this point forward, the spirit continues to exist in an embodied state.
Did I manage to shed any light on this, or just further confuse you?