You are describing a bladder tank, rather than a conventional tank. In a conventional hydropneumatic tank water is pumped into a tank containing air. The air then compresses pressurizing the water. Unfortunately air is soluble in water, which decreases the volume of air trapped in the tank over time. That means that conventional tanks must be somewhat oversized to slow the loss of pressure and occasionally recharged.
Placing a flexible bladder between the air and water prevents the air from dissolving into the water. The air side of the bladder must be "pre charged" to a pressure above atmospheric pressure to prevent stress on the bladder though. It's Boyle's law all over again. Pressurizing the water side to say 60psi would really distort the bladder unless there's a substantial pressure on the opposite side counteracting it.
It's the pre-charge pressure, P1, which is actually the "third" pressure in your equation. It's introduced by the fact that the tank is higher than atmospheric pressure to begin with. In a physics textbook question on Boyle's Law the bladder may be absent and the tank start at a standard atmosphere making the atmospheric pressure "cancel out."
Technically the manufacturer's calculations will be based on the ISA, or international standard atmosphere, and your actual drawdown will vary with the actual atmospheric pressure so it will vary slightly. A given tank would have a slightly higher drawdown in Mexico City since there's less air pressure at the faucet due to the elevation. As a practical matter holding tanks aren't sized to the drop so it doesn't matter.
Put another way any water coming out of the tank must make it's way against the current atmospheric pressure. Typically you would assume ISA to keep things simple since the daily fluctuations are small. The pre-charge pressure, is large enough comparatively to have a significant effect on the drawdown.