Below is a pic of a risky situation. It's called a false grip, or a suicide grip. Don't do it.
Years ago, I was spotting a friend on the bench press. He was a big guy. He was warming up with weight that was well within his wheelhouse. If I remember right, he was using about 185 lbs on his third warm-up set. Jeff weighed in at a robust 260, or so and he was STRONG for his size. No slouch on the bench. 185 lbs is a serious bench for some guys. It wasn't for Jeff. It was child's play.
Jeff was a big proponent of the false grip (pictured below). He was taught by someone "in the know" that wrapping thumbs around the bar was a fatiguing factor and doing so would cause him to squeeze the bar while bench pressing. Squeezing the bar is thought to drain energy from you and decrease your bench pressing performance. Thus weakening you and preventing you from becoming JACKED and SWOLE! It's a fate worse than death in some guys' minds.
While I was standing there spotting him (I was right there), he had an accident caused by using the "false grip." Here (right) is a video of almost exactly what happened to my friend Jeff.
It was a serious accident. He broke his sternum and several ribs. He lost several months of training while recovering.
He was VERY lucky.
What if he had dropped it on his neck. Or on his head?
Those of you who go OH with a false grip, please think about this, because if the bar rolls out of your hands while you have a heavy bar OH - your neck and/or head is likely where it is going to fall.
Please wrap your thumbs around the bar. They call it the "safety grip" for a reason. Any (supposed) meager strength gains you might experience by using a false grip will NEVER out-weigh the very real risk of injury and prolonged recovery of an accident that might be caused by using it.