All good information, and accurate. But' it leaves out one major reason the coolant surges out of the radiator fill point with no cap present, and what the significance of that is..... cavitation and its properties. Cavitation is mitigated by pressurizing the system.
When I service a cooling system and there is still air present, I can fill the radiator fill point full, install the cap, and in some instances fill the overflow reservoir up high. I run the engine until the temp gauge reaches between 1/2 and 2/3. I shut the engine off, don't touch anything, and go away and let the cooling system cool off completely. Invariably, when I return, the reservoir has gone down considerably, and sometimes the radiator has sucked it all back in. I repeat this process as many times as it takes until the level doesn't change in the reservoir or the radiator stays full or both. I let it cool completely and remove the cap and top off or add to the full mark, if necessary.
As the cooling system builds pressure, the air compresses, and, as physics has it, that air is now a much more compact mass and will seek top. That air will get out through a properly working pressure cap valve, and as the cooling system cools off, it will suck liquid back in. Patience is required, along with working knowledge of how these dynamics work together.
I just now performed replacement of a thermostat in a 2008 4Runner V-8. I did the process as I just described. Works like a charm.
Another process is to pressurize the cooling system as the car warms up, to speed up the process, but it is risky and messy, and I do not do it any more. I have time for the patient, safest way.