You really need to ‘try’ to educate yourself on the difference between being born a citizen of the USA and being a ‘natural born citizen’. Only a natural born citizen is, by virtue of the US Constitution, eligible to become President or Vice President. Natural born citizens have two parents that are BOTH US citizens, which disqualifies Barack Obama ....obviously.
And yes....if your baby comes early in the Bahamas AND if you and your husband are BOTH US citizens at the time of that birth, your baby would be eligible to become President or Vice President of the USA some day. If not, your baby would merely be deemed a US citizen, ineligible for the highest and second highest office in the land.
You are incorrect. Child would have to be born in the US. He or she would become a naturalized citizen by virtue of having citizen parents.