If we have two fluids $1$ on top of $2$, I know that the absolute pressure of a fluid $2$ is $p_2 = p_1 + \rho gh$ where $h$ is the height of the second fluid, and $p_1$ is the absolute pressure at the bottom of fluid $1$. In other words, we add the pressures.
Now, consider a thin closed off pipe filled with water as shown, such that the Rayleigh-Taylor instability does not apply:
However, looking at the drawing, why would the absolute pressure at $P_1$ be $P_1=p_0 + \rho gh$ and not $2p_0 + 2 \rho gH$, and similarly, why is $P_2 = p_0+2\rho gh$ and not $2p_0 + 2\rho gh$.
Why don't we add atmospheric pressure the same we do pressure from other liquids?