This method is meant to manipulate the curve after it has been drawn. I would like to be able to grab the curve and not the bezier handle because that keeps the angle of the handles from changing. Doing it that way means the backside of the curve stays the same. Most times it is important to keep the rest of the curve the same and only manipulate that one section.

Unfortunately I cannot record anything on my screen (prohibited at my workplace) but I can describe my process:
1. pen tool > click to set first point
2. move mouse down some distance, shift-click to set corner point directly below first point
3. move mouse up and right some distance, click-drag to make curve anchor

Alternatively:
1. pen tool > click-drag, slightly down-left to make curve anchor
2. move mouse down and left some distance, single-click to set corner point
3. move mouse up some distance, shift-click to set last point directly above corner

Any adjustment of the curve using the bezier handles per the Tim's gif result in his "preferred" behavior, not the "present default" behavior.

Edit:
Perhaps I misunderstood the method of manipulating the curve rather than method of drawing the curve.
Editing/manipulating the bezier handle has the effect I noted as Tim's "preferred", but using the direct selection tool to click and drag on the curve segment itself *does* result in his "present default" behavior.

This is not a normal method of manipulation that I use personally so it took some experimenting to reproduce.
I think Tim has a valid suggestion.

Lance, not sure. Can you demo how you draw the same shape as Tim which then does gets modified in the 'preferred' way?

Does this behavior depend on how you drew the arc with corner?

I cannot replicate the "present default" behavior. Mine behaves as your "Preferred update" does.

True, a solid observation, voted.
Also, thank you so much for the GIF, because these are 1) small, 2) clear, 3) animated. I wish more these clear requests come.