I've been doing bent over rows and dumbbell military presses (if that's the right name for the straight up press) with my hands oriented as if I were holding a barbell. Is there any advantage to rotating so the underside of my wrists face inward?
The main advantage is if you have elbow pain when you're doing rowing movements. Neutral rows (hands held as you describe in your last example) are less stressful to the elbows than pronated (barbell style) rows. I'm not sure if it makes as big a difference during pressing movements.