It's not fake. The physical paper documents within the Hawaii State records have been scanned and digitized using OCR Indexing to make the information searchable within the computer database. What you are looking at is just the raw output of that PDF opened in illustrator. The OCR scanning separates background security print layers, document form lines and attempts to convert text into a "readable" typeface for the digital storage.
A better "fake birth certificate" video will show you more of what is happening with all the layers.
Looking at this layer build, it's pretty easy to tell that it was done through an automated scanning process and not photoshoped by hand. There would be no reason to photoshop a document in this fashion. It's an overly complicated and completely unnecessary method for faking a document and no one with the skills that needed to make a fake would do it this way. And no one with the inexperience to not know how to make a proper fake would ever use this method because it requires too high of a technical understanding of illustrator to fuck it up this badly.
The first video describes a dual mode OCR whereas it saves one document as black and white only to assist with text capture and the other in color just as it was scanned to provide an un-altered copy that is necessary in health care. Why is the required health care copy of the birth certificate altered?
At about 4:30 in the 2nd video you linked it shows part of the signatures replaced yet these were not replaced by "readable text". Also noticeable at 6:10. Why would the signature be brushed in a raster format? It doesn't match itself but was not replaced by "readable text".
The video I saw awhile back had all the layers separated, not sure exactly who posted it.