I am confused...my in-laws have left the US with their Dominican passports; my son who has dual citizenship has left the US numerous times with his Dominican passport. Also, the passport you present at check-in is the passport information that is put into the airlines system. You are a US citizen (born/naturalized) whatever soil you are on.
...And it is easy to get confused as the rules vary greatly from country to country and whether you arrive by air, land or sea.
Suffice to say you need a US passport when entering the US from the DR, or a Dominican passport with a Visa stamp, or a Dominican Passport along with a US permanent resident green card or a special waiver. It matters not what your DR passport indicates as your place of birth...it is totally irrelevant for admissibility documentation purposes.
You are a US citizen if born in the US...unless you have renounced your US citizenship, in which case you will need a foreign passport with a visa......unless coming from Canada or Mexico or other countries not requiring a visa to enter the US. Speaking of which, if you are a US citizen you may enter/exit the US by land with a passport card.....no passport needed.
And then there are all the requirements/rules for cruises, closed/open loop.
And then you can leave the US without a passport from any country (Voluntary departure) with DHS permission. (this is for those who entered illegally do not have a passport and wish to return home.) Easy to leave.....harder to get in.
There are lots and lots of rule specific exceptions.
Respectfully,
Playacaribe2