For what it may be worth -- as a beginning Christian Harmony
singer, in the context of beating time, I was taught two points:
1) whatever you do, as leader your obligation is to be clear, and
2) the hand goes up and over the bar line, and down on the other
side.
My impression is that it's usually called "the downbeat", and mostly
for a reason.
Dan Huger
Asheville, NC