The shot from the fairway stopped short of the first even though it landed in the same spot. That’s because you generated much more backspin due to the cleaner lie. And for the teed-up third ball which had zero grass on the clubface to interfere with contact you created max backspin and stopped the ball almost immediately after it hit the green.
It’s not all purgatory. Many of Shinnecock’s greens are downright friendly with raised edges that funnel shots toward the center of the putting surface. Most however are shaped to repel shots away from the flagstick and in some cases off the green entirely. Be advised: every green features serious undulation. A few are so sloped that it’s impossible to imagine the ball stopping on its own.
Whenever you set up for this shot be sure to make a few practice swings to check the position of your scuff marks on the ground. At address you want your ball to be centered in the area that your clubhead normally scuffs which will give you the cleanest contact possible through impact. It’s not all automatic however — it’s still up to you to figure out how the ball will break and roll after it hits the green.