I hold both at different times. I hold the upward bend and pull back when I am torquing the pedals. If I am cruising steady state on a flat section I just rest my hand on the flat part near the shifters.

For me, depends on the road surface and it's pitch. On smoother, flatter surfaces I like to get a little more aero and grip the extended flat part, but if you are new to this angle on an aerobar, your wrists may be a little sore for a while. I actually find this to be quite comfortable now. On rougher roads or when I am climbing, I grip the "bend" part and tend to torque on the bars for a little more control/power.

I ride on the flat part of the bar many times without even griping around the bottom. I just put my palms down on the top. It doesn't put any strain on the wrists.

Cool. Just wondering what is the most "aero" way to hold the bars

You are right, I find myself doing the same thing a lot too. However, one time I was just laying my hands flat on bars and hit a dip in the road and almost biffed it hard. I at least like to have a little bit of a grip on the bar, not too tight, but some grip just in case. It also helps when idiot drivers come up behind you and try to play tag with their rear-view mirrors. The flat part of the s-bend is a lot more exaggerated than most are used to and it took about week or so just to get used to maintaining my hand/wrist angle that way. No problems with it now.

The most aero way to hold the bars is the way that keeps your forearms level. If your elbows are higer than your hands this will not only have more arm surface to the wind, but the higher elbows indicate a higher torso position too.

Height of head/torso makes the difference more so than hand position, although they are related. Whatever position that leads you to "sit-up" more will be less aero...

I hold them more forward on the flats when I ride on more level roads, and I hold them shorter on the upward section when I do rolling hills or short power climbs.
Hope this helps.

Herbert
Litespeed / Quintana Roo