Talking about different cultures, let me tell you how it works here at south of the Equator....
In my country, it's very common for couples to want to registry birth in video. Most of the best hospitals have their own professionals for that, and of course it's done only if there's an informed consent signed.They are professionals;they never go into the room before the patient is totally covered, anesthesia done and catheterizing as well ( if it's the case), they only record the baby, avoiding to record women's genitals, and then they edit the video to make it ready and appropriate to be seen by all family. They put on music, the names of parents and doctor, and so on.It's all done with the biggest respect to the mother and the child.
They do live transmition of the birth to a TV intern circuit, so that the family can watch it during the birth, if they want.
They are trained to avoid bad images, and they know when it's to stop transmition, if anything goes wrong.
It's a great work, and a great gift for the child when he grows up....When my children were born it wasn't possible to have this done, I only have pics taken by my husband, and as he was a little nervous, you can picture how good they are...LOL. I wish I could have some professional ones now.
Nina, from Brazil.