Those are the main ways, there may be more. In any event, most countries (the USA is actually an exception to this rule) DO NOT recognize place of birth as an adequate means of conferring citizen ship. If you are merely born in France, Italy, Switzerland, Germany, Mexico, Brazil, China, and most other countries, you do not become a citizen of that country by that alone. The USA does recognize this manner, and that's why many pregnant Mexican women illegally cross our border to bear their kids on US soil - they immediately become citizens. Despite the public debate over the US/Mexican border and illegal immigration, the USA is actually one of the easiest places in the world to get citizenship. Historically, this is because the USA has wanted to grow their population quickly, while most other countries don't. France doesn't care if you are born in Paris, you are not French because of that, and they are in no hurry to add more French people to their country, or to the planet.
If one (or both) your parents are from a country, then you are a citizen of that country. Almost every country on earth recognizes this policy, and I can't think of an exception. There have been some historical limitations on this, such as, for example, citizenship could only be conferred through one parent (usually father) but not both. More recently, many countries are allowing both parents to confer citizenship, for example, Switzerland as of 20 years ago now allows children to gain citizenship through their mothers, which previously was not allowed (Swiss women just got the right to vote in 1974 also, by the way). So for example, two Americans while on vacation bear a child in Paris (or Rome or Mexico City, etc.), that child is an American citizen only (based on parentage, as the other countries don't recognize place of birth). However, if two French, Italian, Mexican, etc. while on vacation bear a child in Los Angeles, that child becomes dual national - by place of birth according to US law, and by parentage according to the other country's (regardless of wheter our law is the same, which in this case, it is) law.
Regarding dual (or triple, quadruple) nationals, most countries recognize this (Including USA), and it's OK. For a long time, the official US policy was to discourage dual nationality, however, there are now so many dual nationals in the US that the US policy has changed to "it doesn't really matter anymore."
A simple google search will add more info, or just visit the US State Department Immigration and Naturalization website, it should all be there.
As far as ANS and her child, Danielynn is definitely American, because ANS was American. Whether the child is Bahamian depends on whether ANS was merely a resident or a citizen when child was born (I think resident only, thus does not confer citizenship), and also whether the Bahamas recognizes place of birth as conferring citizenship (most countries don't).
But regardless of citizenship, which courts have jurisdiction is another matter. If the baby is in the Bahamas, the Bahamian family court probably has a claim on jurisdiction, although US courts might claim it as well.
As far as who should get custody, it's easy enough to determine parentage with a DNA sample from both parents (to prevent child swapping, because a swap is possible without ANS' DNA). As the child is very young, she hasn't bonded with anyone yet (no in loco parentis issue here), so as soon as paternity is determined, Dad gets custody.
That's my guess.
But the more interesting question, is of course, where is the $$$ ?
And is baby Danielynn flat broke, or a multi-millionaire? that Q is already before the Supreme Court of USA, though not exactly in that form. Basically, the question is, who has jurisdiciton over the probate issue of multi-millionaire Marshall's estate, the TX court, or the Fed. Courts? If TX wins, ANS' baby gets zero. If Fed wins, ANS' baby hits jackpot. Since the Feds are deciding the question, I'm leaning toward jackpot, but anything can happen.
Good Luck.