The article you link disagrees with you.

The chain reaction can be shut down in a matter of seconds. However, the reactor continues to generate a huge amount of heat for weeks afterwards. It's not as much heat as a fully-running chain reaction, but it's still enough to cause a catastrophe if cooling is not kept running - we're talking multiple megawatts of heating here. The cooling requires an external power supply. It was the failure of this external power supply that caused the problems at Fukushima.

There are newer reactor designs that are specifically engineered to be capable of cooling themselves on shutdown without an external power supply. However, most reactors in the world do not have that capability.

Your link supports the comment you are replying to; it says that after the reactor is shutdown, additional cooling is needed for some time, and that eg the Fukushima and Three Mile Island incidents both involved damage done by an inability to cool the reactors after they were already shut down.

No they are technically correct. In the reactor world "shut down" explicitly and specifically refers to chain-reaction-running mode. The reactors do still require active cooling for several days+ to avoid plant damage scenarios. Several comments on this page seem to not have a clear understanding of this difference.

Once the chain reaction is shut down - there is basically no way for the reactor to come alive (chain-reaction-wise) again on its own - not even if it's being bombed, shelled, etc. And run-away super(prompt)critical reactions are not even possible with this reactor design. These reactors are water-moderated - which means that water is used to slow down neutrons to increase their reaction probability. As the reactor heats up, the water gets less dense (even if it is still a liquid) making it a less effective moderator - this density decrease is enough to passively/automatically keep the reactor in a shut-down state. Residual decay heat from radioactivity of the fission byproducts post-shutdown is enough to damage the reactor internals for several days - hence the need for active cooling post-shutdown.

I've seen lots of crazy-exaggerated news reporting on how "bad" or "dangerous" this entire situation could be. It's not good, but neither is the war in Ukraine. It's not even remotely possible for this to be anything like Chernobyl, and I think unlikely to be nearly as bad as Fukushima (which in the grand scheme of the Tsunami - wasn't really that bad). In the war context, I don't think this nuclear plant situation is particularly notable beyond it providing a large fraction of Ukraine's power.

I am a nuclear engineer FWIW.