As you have guessed it is not a killdeer. I have come across them and my daughter Jennifer has been dealing with one at work. They are ground nesters. The one in the below photo's with the stripes is the killdeer I found at the Bridgeport Cemetery. It was not one bit afraid of me just like the one my daughter dealt with.
I suspect Scott may still be correct. Since I have the full size video I was able to go frame by frame and enlarge it and the beak is long. I tied to get the best views but that was hard.
The beak is long.
Compare them. The eye on the one in the nest looks exactly like the eye I see online for the woodcock. See how thick the beak is by it's head. A sure sign of a long one.
The Killdeer in the Bridgeport cemetery and an egg.
With that all said, the bird that was faking it seemed bigger, more like a dove that Gibby mentions. But it was opening it's wings trying to fake me out.