I've played around with using a short thumb on the top hand when I use short clubs (scoring irons and wedges) but have always relaxed the low hand thumb on the side of the grip. I still make the adjustment on occasions, but tend to just stay with my standard set up.
I did not like the short thumb idea on longer irons and woods, it seemed to cause to much grip pressure.
I see many players who want to press the low thumb into the grip, which I try to explain to them it will cause tension in their grip.
I think pressing the low thumb into the grip leads to a player to slice their shots.
I think players who use a strong grip will benefit with a short upper hand thumb.
Sometimes I will have a little soreness in my left thumb (top hand) which really plays havoc swinging the driver, and then pull the thumb in to protect it.
This past September, both thumbs have been tender along with soreness in my wrist.
Had to increase the Advil dose during this period of time, just to enable to manage the pain for playing.