According to Pascal's law, the pressure affects all directions, so I wouldn't make any distinction between horizontal and vertical pressure. The change (in hydrostatics) is only given by your $\rho gh $ formula.
Okay, more pressure should imply more temperature, but this temeprature change is completely negligible for most practical cases. Just take a book and press it against two other books (all of them at the same temperature). Then measure temperature again. You won't find any significant change, for sure.
That's because you're producing a very macroscopic force that creates a small elastic deformation. Once you stop making the force, the material liberates and gets to the same original shape. This is barely trasnferred to kinetic energy of the single molecules. You would have to somehow excite normal modes of vibration if you want to heat up the material. That's how a microwave works.
The quoted paragraph from the textbook talks about fluids which usually includes gases, liquids, and plasmas. However, it would not be right to say that for liquids (e.g., consider water for concreteness) the pressure is the kinetic pressure $P_k=nkT$. First of all, we know that we can put water under a piston and increase the pressure isothermically at nearly constant density. If the pressure is due to particle collisions then why does it increase without any increase of temperature and density? Furthermore, using the numbers for water at normal conditions, $n=33e27 m^{-3}$, T=300 K, we'd get the kinetic pressure $P_k$ at about 10 million atmospheres, but we don’t see it!
We don't see this huge pressure because it is largely compensated by intermolecular attraction forces. So the total pressure in a liquid is $P = P_k + P_f$, where $P_f$ (negative at normal conditions) is the component of the pressure due to intermolecular forces, strongly dependent on the density. If water is compressed (at a constant temperature) the resulting pressure increase is due to the change of $P_f$.
So, for water compressed under a piston at a constant temperature, the total observed pressure increases; the thermal pressure caused by water molecules bouncing off the surface does not change in this process but the intermolecular forces respond to the compression changing the total pressure.
Given that the thermal pressure in a liquid is almost entirely compensated by the intermolecular forces, one can model a liquid as a large number of slippery almost incompressible balls lumped together, essentially excluding thermal motion from the picture. This model would have the properties of a real fluid (weakly compressible, isotropic pressure, Pascal law, Archimedes law). If we put such a "liquid" in a vertical column then we'd observe that those balls deeper down from the surface are compressed more (because there is a larger weight above them), and a body embedded in this “liquid” deeper would experience a larger external pressure.
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Actually they both work in the same manner. The cause is the presence of gravity.
Pressure increases with depth in a liquid, because the heavy (dense) liquid has to carry the whole column of liquid above it. A water particle at the bottom of the sea must hold up all the water above it and all the air above that. The water particle at the surface only has to hold up the air above it (corresponds to standard atmospheric pressure).
It is the same thing for air and other gases. And as you might already know, the atmospheric pressure at ground level is much bigger than the atmospheric pressure at an air plane in a height of 10 km. Just watch any aircraft crash movie and see how everything is suched out when there is a breach because of the lower outside pressure...
For gas within an earth sized container, the pressure difference because of depth is so small because of the very low density that it simply doesn't have to be considered.
Yes, in outer space where no force like gravity pulls all particles in one single direction so they have to "carry" reach other. But in that case it would also be difficult to define depth...
I should mention though that such liquid in outer space of course exerts it's own gravitational pull. If you have large quantities of liquid (or gas for that matter - just look at a gas planet), and I mean very large quantities, then the liquid will form a sphere and the pressure will increase as you dive deeper. But this depth is then measured towards the center of this sphere.