The line between a pure science question and a worldbuilding question and answer is sometimes very fine indeed. In worldbuilding, context and background are important. Something is done 'in a world'.
Let's start with pure science.
Take a large sealed drum, on its end. Within that drum, siting upright on the bottom, is a much smaller diameter cylinder, Cylinder A, open at the top. We will call inside this small cylinder Region A, and outside this cylinder Region B. Region A has some earth on the bottom, to Level A with respect to the bottom. Region B has water also to Level A. The pressure at the surface of the water and the earth is the same.
Now, inject more water into Region B without breaking the seal so that the water level rises. This decreases the volume of air, so the air pressure goes up. The pressure at the surface of the water AND the surface of the earth is higher, even though the level of the earth did not rise. This pressure rise is entirely due to the compression of the air in the entire sealed drum.
Same setup as scenario 1, but now add a second cylinder, Cylinder B, open at the top, in Region B. It is filled to the top with water, at a level much higher that the water in Region B. The pressure at the surface of the water in Region A and of the surface of the earth is the same.
Now, open a stopper in cylinder B so that the water in it flows into Region B. The water level in region B rises considerably, much higher than the level of the earth in Region A, but no new water is added. The existing volume of water is just moved around and redistributed. The total area for the air is unchanged. It is not compressed, and so the pressure does not change at the surface of the water or the earth. It is important to know if the additional water came from within the system, or from without.
Same as Scenario 1, but the drum is not sealed at the top.
Now when more water is added to Region B, the water level in this region rises, but the air is just pushed out the top, the air is not pressurized. There is no change in air pressure at the surface of the water, nor at the surface of the earth. There is no compression. There is no additional pressure. It is important to know if the drum is sealed, or open so that the air can escape (into space?).
Same as scenario 1, except that the smaller Cylinder A goes right to the top of the drum, and Region A is sealed separately from Region B, as well as the drum being sealed.
Now, when water is added to Region B to raise the water level, only the pressure in Region B goes up. The pressure in Region A is unchanged. The pressure at the surface of the earth does not change.
The first four scenarios ignore gravity to a great extent, and thus the weight of the air causing pressure on the surfaces of the water and the earth. So this scenario is like Scenario 3, only let's add 'weight' to the water. In this scenario, the drum is VERY big, and is in a vacuum, but siting upright in a gravitational field. Gravity prevents the air from spilling over the top of the drum. In this scenario, Cylinder A goes to the top of the open drum. There is just enough 'air' in Region A and Region B to fill them up half way. Thus, air can not go between Region A and Region B ('over the top of the wall', as it were).
Adding water to Region B to raise the water level 'pushes' the air up in Region B, but does not compress it. The air is unconstrained, and can move up freely and unrestricted. The water level goes up higher than the level of the earth in Region A, but there is no effect on the volume or mass of air in Region A, and thus no pressure change at the earth level. There is only a very slight pressure decrease on the water, because now the air is further away from the center of gravity, so there is less 'pull' on it. Whether the additional gravity from the additional water compensates, is moot. There is no change in Region A.
Same as Scenario 5, except that Cylinder A does not go all the way to the top, and there is enough air in the drum such that the air at the top of the drum can flow over the top of the cylinder between regions A and B. The top of the drum is still not enclosed, so the volume of air is unconstrained. The top of the air does not reach the top of the drum, so it cannot flow out of the drum.
Now, when water is added to Region B from outside the system to raise the level, the water displaces air further up in the drum. The pressure of the air does not change, as it is not constrained and there is no compression. The volume the air is contained in does not change, it just moves higher up in the drum. The top level of the air goes further up the drum, and can 'spill' over the top into Region A. There is no overall pressure change of the air, as it is not compressed. But now there is more air in the air column over Cylinder A (the height of the top level of the air has gone up) so there is more mass of air on the surface of the earth in Region A. The additional 'weight' due to gravity of the air in Cylinder A and Region A causes an increase in pressure at the level of the earth. Thus, it is important to know what the height of the atmosphere is, with respect to the height of the wall.
Also, considering Scenario 5 with this scenario, is there free movement of the air at the top of the wall between Region A and Region B? Are there weather patterns that prevent this? Like inversions, for instance. Is the air pressure already higher over the land than the water, due to other variables (local heating, atmospheric composition, air turbulence), that causes a local high pressure system preventing air movement from Region B into Region A?
Same as Scenario 6, but now Cylinder B is added inside Region B, that holds the water that will flow into Region B (a frozen glacier that melts?).
As the water flows out of Cylinder B and into Region B, the water level in region B goes up, but the overall volume available for the air does not change. Air is not 'pushed up' in Region B, it just fills in the space where the water in Cylinder B came from. The overall height of the air in the drum does not change, and thus there is no change in height of the air in Region A, and thus no pressure change at the surface of the earth. It is important to know how the water that is used to raise the level of the water in Region B affects the overall volume available in the drum. Is the water just displaced from somewhere else, and thus the overall volume remains unaffected?
This is similar to the common misconception over melting icebergs causing the sea level to go up. If the iceberg is already floating in the water, its melting will not significantly 'add water to and raise the level of' the sea level.
Same as Scenario 6, only there is a huge pool of hydrogen at the bottom of the drum, below the water in Region B and the earth in level A. The air is composed of oxygen.
This pool of hydrogen is released into the air in Region B, combines with the oxygen, and forms water. This water fills up Region B, and raises the water level. But this water is just replacing the volume taken up by the oxygen. In this scenario, the volume of air goes down, perhaps more than the volume of water goes up. So the air pressure actually decreases in the system. The air pressure over the water AND the earth goes down. It is important to know where the water comes from.
A bit of trivia, this was once the theory of where water originally came from on earth - hydrogen from volcanoes mixing with oxygen in the atmosphere - until the 'water from comets' theory gained dominance.
Same as scenario 6, only the atmosphere is made of a gas that is very highly soluble in water. Ammonia, for instance.
Thus, when water is added to region B, the air is dissolved into the added water, the volume of air goes down, perhaps lower than it started out to be, and thus perhaps the overall height of the top of the atmosphere goes down. It is important to know the effects of the added water on the existing air.
Same as scenario 6, only the air goes to the top of the drum.
When water is added to raise the water level, the air is pushed out of the drum completely and into space. Atmosphere is lost, as was the case on Mars. Thus, it is indeterminable as to what happens to the column of air over Region A, and thus the gravity-created pressure on the surface of the earth. It is important to know what happens to the air that is 'pushed up' or displaced by the rising water. Is it still constrained by gravity to the planet? If the water suddenly filled Region B with a 'whoosh', it is more than likely that much of the air was just 'blown off' the planet.
In worldbuilding, as opposed to a science lab demonstration, it is important to take into consideration the entire context of the world, and not to assume the perfect science lab conditions and control of all of the variables. The real world is very different from a contrived and proscribed science demonstration, and is full of 'uncontrolled unintended consequences'. Thus, taking a purely science approach to a worldbuilding question could lead to incorrect answers, that work perfectly in the laboratory but fail completely in the context of the world.
There is just not sufficient content, context, and criteria in the original question to assume only scenario 6 under the perfect atmospheric and climatic conditions applies. This question begs that the answer approaches the question from many perspectives, not just the purely laboratory response.