Yes, that's a more correct understanding. The person is standing as a proxy. The prayer for the baptism of the dead is exactly like the prayer for the living (as used in many churches including the LDS), but includes "I baptize you for and in behalf of ___ who is dead" after calling your name and ending with "in the name of the father, the son and the holy ghost, amen."
That last sentence is right. After you die, you still have your free agency to act however you want. We can still make choices. The gift of choice can never be taken away from us. We will always be able to think for ourselves. If someone has not had the oppurtunity to hear of Christ, but must to be saved, we believe that they are taught in heaven. Afterwards, they have the choice to make of whether to accept it or not. If they choose to accept it and are baptized for on earth, they can progress through the kingdoms of heaven (Celestial = top, Terrestrial = middle, Telestial = lower). Now, I know you're going to challenge that statement, but think about it, if you were absolutely perfect in your life and did everything right, would you want to live with someone who could barely sqeek by into heaven? or vice versa? Because we still have the agency to choose, people will still choose not to believe, just as people still don't believe here on earth. And, before you ask, you're not taught by God or Jesus. You're taught by the faithful who believed on earth and are asked to become teachers.
There's a lot more I could write about that, but let me sum it up by saying this: when you die, you don't automatically go to Heaven or Hell. You go to a type of "holding ground" per se, where you wait to be judged. This is where you are taught. God and Jesus are not physically here. Then, you are judged, and go to either heaven or hell, depending on how you are judged. If you accept, you have to have all the proper work done on earth before you can go all the way to the highest tier. That is why we do work for dead.
H-D