After the second round of the Barclays, Phil Mickleson switched to a modified claw grip when a friend of his told him about my modified claw grip. Phil has always been an arc putter. The claw grip produces a PILS (Pure In Line Stroke). I believe a PILS stroke is more consistent and accurate.
The original putter grip was a baseball grip because greens were not as smooth or as well manicured as today. However you lose feel and touch.
As greens became faster and became better maintained, the interlock, overlap and reverse overlap came along. Golfers gained feel and touch but became wristy.
The claw was born to eliminate wrists and create consistency. There are three variations of the claw. First, allow the shaft to rest between the fore finger and thumb. Point the fingers and thumb toward the hole. Second was using the same grip but wrap the fingers around the putter shaft. Last was to point the fingers down the shaft either at an angle or straight down toward the putting surface. This eliminates any wrists and forces the shoulders to rock back and forth creating a PILS stroke.
Last grip to evolve was left hand low. Left hand low produces a great putt stroke on short putts but creates distance control issues.
Before selecting a putting grip, you need to determine your strengths and weaknesses then select a grip that creates the most consistent and accurate putt stroke for you. This is best done under the instruction a professional putting coach.