Water Bottle Rocket Thrust - two calculation methods not matching

I have been launching water bottle rockets with my kids and we are in the process of creating a simulation of the launch using numerical methods. I am a mechanical engineer, but fluid dynamics is not my strong suit.

Can you give me an intuitive explanation why the two methods below give a factor of 2.0 difference for thrust? Which method is correct?

First Method

I found these formulas for calculating thrust at the following website. https://www.ohio.edu/mechanical/programming/rocket/analysis1.html

Thrust equals mass flow rate times the exhaust velocity.

$F=\dot{m}v $

Mass flow rate is found using the water density, nozzle area, and exhaust velocity.

$\dot{m}=\rho A v$

Combining we get

$F=\rho A v^2$

The square of the exhaust velocity is equal to 2 times the internal gauge pressure divided by the water density.

$v^2=2P/\rho$

Combining we see that the thrust is equal to 2 times the nozzle area times the pressure.

$F=2AP$

Second Method

In the image below, it seems that the thrust on the rocket should be equal to the force imbalance as shown by the arrows. This force imbalance is the internal gage pressure times the nozzle area.

$F=AP$