There still seems to be a terminology problem. Below is a quote from the Hawaii Department of Home Lands. It calls the long form "the original Certificate of Live Birth." They call the computer generated form the Certification of Live Birth.
In order to process your application, DHHL utilizes information that is found only on the original Certificate of Live Birth, which is either black or green. This is a more complete record of your birth than the Certification of Live Birth (a computer-generated printout).
Your are correct.
I have previously read where it was simply called a birth certificate rather than a certificate of live birth.
Unfortunately, when we read different posts or sources, we see differnt terminology.
I do have a picture of the desired document which does say Certificate of Live Birth.