Are there two ways to finish a rear naked choke..

Eg let's say my right arm is the choking arm and goes in front of their neck, and my left arm is the supporting arm and goes behind their neck.

One way to finish is a rock the baby like motion with my arms, going to the right, so in the direction of my choking arm's elbow

And the other way is more like a shrug motion so instead of my arms sliding right and up, they might go right a bit but they go mainly just up.

Is one of the methods an older way, and one a newer way?

I've heard one or perhaps both of these methods described as like slitting the throat cos like the left thumb runs close past the side of their neck.