I have the exact same type as Trump. I, too, was born in NYC. That IS the original. No other form exists. Not with the NYCDH or any other City agency. It is accepted as legal proof in a court of law. All records are on microfiche anyway, so if you lose the original, you just get a copy of the record.
I have a hospital certificate like Trump’s, too. It’s quite nice, but it is not the legal government-issued (numbered, with seal and signature) that is required for most official transactions requiring proof of identity.
Many people have two, both the hospital paper and the official state-issued certificate that includes all the birth details reported to the Dept. of Health by the hospital.