Apparently, Lincoln overcome with emotion added "under God" as he spoke. That's considered the official version because that's what he said there. Lincoln, unlike BO, didn't have a problem with God. Lincoln, not unlike many people, had a problem with organized religion but considered himself a God-fearing man.
Now what would be the point of having the prez read a draft? That's stupid. Did the others read drafts? No, they did not.
The president was the only person to tape a version, known as the 'Nicolay' version, which perhaps most notably is one that does not include "under God" in the second to last sentence. The selection came at the request of the documentarian Ken Burns, who has a team heading up the project. One of the producers told CNN they gave Obama this version to use to highlight that there are multiple versions.
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