Another scenario is when sometimes the person catching the baby will announce, "Hold on, the cord's around the neck." If the parents don't understand what that means they might think something's wrong with the birth, but this actually very common not a complication. Hopefully, the health care provider will explain that are just pulling on the cord a little to see if it's very tight. If it's not tight they'll loop it over baby's head, but even that's not necessary. Remember, baby is bringing one end of the cord with him, so it's usually not holding him back. If it's very tight the provider will quickly clamp and cut the cord, but even very tight cords do not hurt the baby.
Folks from my generation might remember the E.R. episode where Dr. Carter's baby suddenly died a few weeks before the due date because the cord got a knot in it. This is a tragic event, but is not the same as what happens when the cord is around the neck. The umbilical cord is made to resist kinks and knots that will restrict blood flow. Ultimately, the things that can go wrong with the cord like a true very tight knot are not preventable at all, but they aren't common, either. So relax, drink some water, eat a healthy snack, and only worry about the things you can actually do something about.
To understand all you need to know about how the cord works and the research showing how common and harmless it is please read this article from Bellybelly.