There is something deeply wrong with how I am trying to understand this. I know that the surface of the water exposed to the atmosphere doesn't matter because I have read it in multiple sources. But I don't understand why.
When I look at the picture that you posted at the top, I see that there is a certain amount of surface area of water in the petri dish that is exposed to atmospheric pressure. So, the equation pressure = force/area leads me to believe that force = area*pressure which seems to mean that the more surface there is the more force will be applied to the water.
Or as an example, if I apply x pressure to a 2m^2 side of a rock, then the force applied = 2m^2*x. But if I apply x pressure to a different side of the rock that is 3m^2, the force is greater f = 3m^2*x; this might move the rock where the other force might not.
This is driving me crazy because I know I am wrong, but I feel like I am right.