"negative pressure downstream, dissolved gases coming out" - this is correct, and that's a great video showing it. It's called cavitation in some contexts.
The question of at what pressure this happens has three parts, as far as I understand it.
(1) The local temperature and pressure (i.e. resulting from valve action) need to be such that you are at a supersaturated condition. Vapor pressure of would-be-gas-species > pressure. First and foremost, this depends on the amount of the species in question in solution to start with. There are also considerations of physical chemistry that affect the vapor pressure curve. However this is only a high level description of the equilibrium, you won't actually see bubbles when the temperature and pressure of this condition is reached, at least not instantly. This condition will be roughly good enough to slowly grow bubbles in corners or scratches in the side walls, but the fast phenomenon we are talking about is homogeneous nucleation.
(2) the classical description of bubble nucleation. This brings in the effect of curved liquid-gas interfaces. The effect of surface tension is called Laplace pressure, which increases pressure with curvature and tends to drive the gas back into solution. Vapor pressure is also altered, per the Kelvin equation.
(3) Lastly, classical predictions based on Laplace pressure have contradicted experiments, especially at small but well-above-molecular scales, which drags us deeper into thermodynamics. There may be some unresolved points of theory here such as how surface tension varies, and I am not that solid even on the basics at this level, but you need to be past something called the spinodal point, where the second derivative of Gibbs free energy is zero.
For engineering purposes, our goal is simply to ensure that there won't be any unintentional gas phase, so it is good enough to make calculations based on the equilibrium of vapor pressure, #1 in the list above. The solution is to control some or all of: pressure, temperature, concentration of would-be-gas-species in the fluid, or valve opening speed.