The document specifically states that declared aliens are to be Struck out. Obama is Struck out.
The document is a renewal process, indicating Soebarkah is an Indonesian citizen. Soebarkah is an Indonesian. He may not be getting dropped. He is a Minor child of an American citizen that needs to travel with her, but he is being identified as an Indonesian by being struck out. This is pretty clear.
Yes, she is allowed to AMEND her passport to add or drop children. That would be an AMENDMENT and if she was doing that, she or the passport authorities would have checked the AMENDMENT box on page 1.