No, a parent cannot renounce citizenship on behalf of a minor child. Only renunciations of citizenship by the person after attaining the age of 18 are valid under US law.
The only question open is whether he was a citizen at birth, which because his father was a British subject, and his US citizen mother was younger than the then prevailing statutory requirement for one parent’s US citizenship to be passed to a child at birth, turns entirely on where Obama was born.
You are correct about the place of birth, but the Indonesian question is still open. If he traveled on an Indonesian - or British, or Kenyan - passport that he obtained as an adult, he almost certainly did renounce his American citizenship to obtain it. And at the time of his trip to Pakistan, they did not allow Americans to enter.
He did NOT need to do ANYTHING in America, or provide any notice to America, to renounce his American citizenship. Making an entitled claim of citizenship in a foreign country is probably enough. I could have claimed full British citizenship at age 18, because I was born in London to American parents - and I would have had no further duty or obligation to America of any kind. Instead, I took an oath of citizenship here in the USA, and have a (derivative) Certificate Of Citizenship that attests to that fact, and shows my date of AMERICAN citizenship as the same as my date of birth.