I am not a lawyer. Important to keep that in mind.
I have heard of cases of parents being investigated after an infant dies (during or after birth). I cannot recall ever having heard of a mother being sent to jail for homicide because her baby died being born or soon after. And I can remember cases where women gave birth and abandoned their babies in dumpsters.
I would, however, caution you that videotaping the birth might not be the best move to protect yourself in the event of an investigation. In the US, the criminal justice system places the burden of proof on the accuser. It's not up to the accused to prove that there was no wrongdoing, it's up to the state to prove that there *was*. If there's no tape, there's no evidence, but if the birth is captured on video, the district attorney's office can go through it frame by frame and question everything. The DA may have different opinions than you do concerning when you should have known things, what you should have done, and how fast it should have been possible to act. The location and perspective of the person with the camera can lead to misrepresentation of your experience. And in the event of an emergency, you're going to conscript every pair of hands to act, including the set that's holding the camcorder.
Birth videos can be a lovely memento, but don't look to them as protection from legal action.