“But U.S. law would still recognize Obama as an American citizen”.
“If Obama had Indonesian citizenship for a period, it may not necessarily have changed his U.S. citizenship status, but it could raise loyalty concerns.”
You don't need to look it up. Another country, or your parents, or ANYONE (not even you) can lose your U.S. citizenship while under the age of majority (21). (and in some cases even after that (Perkins v. Elg, 307 U.S. 325 (1939))
In Perkins v. Elg, the court ruled that a person does not automatically lose their naturalized citizenship status merely by moving overseas. But Obama was born here. You can argue that he's a natural born citizen or a native born citizen, but the fact is that he was a citizen at birth. That cannot be taken away from a person, it can only be relinquished voluntarily and intentionally. And according to U.S. law a parent cannot do that for their child.
This applied mostly to immigrant families from Ireland and Italy who were sending the girls to convents abroad to have the kids ~ most of whom would be shuffled into Catholic orphanages and then raised to be part of the religious community.
Not according to the US State Department. Or do you know more about the rules than they do? And in Perkins v Elg, the decision was that she did not lose her citizenship.