Go to City Hall and ask to see the original copy filed with the city. It will be different and have more information.
What you have, if it has the county (or borough in NY City) seal stamped in, is indeed a legal document for things such as passports. Perhaps NYC provided those as a courtesy when you were born. Not all juristiction would do that. But it is not the so-called long form birth certificate which has much greater detail. Two different things.
I don’t know where you get your information, but private hospitals in NYC issued their own birth certificates.
My BC on file at the NYC Dept. of Health does not list any demographic info such as race, religion, etc. I have the original hand inscribed copy. It was sent to my parent’s home address by mail. The copy on file, of which I have copies, lists my name, date and place of birth, and name of Hospital.
There is such a plethora of different kinds of BC in each State, County, hospital, IMHO the Obama BC chase is a wild goose chase and total waste of time. It poses interesting questions, for which there is no legal resolution.
From a legal point of view, no photocopy is accepted as legal evidence in a Court of Law because forgery and “Photoshop” techniques have become so sophisticated, authentication cannot be proven beyond a reasonable doubt. The same is true for photographs.
There exists no “long form BC for me. My family came to America in the 17th & 18th century. Does that mean I can never be POTUS?