The only reason to be uncertain is if you are an illiterate. The State Department documents clearly state why you would write someone's name there and cross through it. It is because that person has forsworn his American nationality and naturalized as a foreigner.
An eight-year old Philip Nolan? That's a bit much.
Do you think children renounced their US citizenship so often that there were specific regulations to deal with that?
I don't know the exact law of the day, but it seems like Barack was too young to renounce his US citizenship himself and his mother couldn't renounce it for him.
The area where the name is written and crossed out was to add or remove a child from the mother's passport. It's not clear whether you're including or excluding a child by writing the name in there.
One possibility is that the mother thought she was adding his name and realized that she'd be removing it (or vice versa), so she crossed it out.
I'm not saying he definitely wasn't adopted and definitely never lost his citizenship. I'm just saying there are other possibilities.