> In the split second it took for the rod to travel the remaining 8.3cm, the reactor spiked to 20 GW, 6300 times its safe operating capacity.
(Up from that days' configured power output of 3 MW, happened while operator was lifting a control rod)
Prompt neutrons are emitted on a very very short timescale and so if the power were controlled solely by them it'd be impossible to control a reactor because any change in the control rod position would cause an extremely rapid change in the power. Instead, reactors are designed to be subcritical considering only prompt neutrons and then be made critical via the remaining delayed neutrons. This extends the reactor period (time it takes for the power to change by a factor of exp(1)) from milliseconds to minutes.
However, if there isn't enough control material in the core that the proportion of prompt neutrons is enough to make the reactor critical by itself then it goes "prompt critical" and exactly what happens in the article occurs. Reactors are supposed to be operated with enough margin that this won't ever happen (which isn't that difficult), but it sounds like they were deliberately operating right on the limit, which scares me.
However, I think you made a mistake (probably a typo but I think it's worth noting it to avoid confusion of other readers) in that sentence: “However, if there isn't enough control material in the core that the proportion of FAST neutrons is enough to make the reactor critical”.
Prompt neutron (short time between fission and neutron emission) should not be mistaken with fast neutron (high kinetic energy, see https://en.wikipedia.org/wiki/Neutron_temperature)
Likely reactors are run with enough margin these days because of the lessons learned at places like this - it was an experimental reactor, the goal of which is to learn stuff.