So, what we colloquially call 'reptile brain' turns out to be somewhat accurate.

There is some truth (but not accuracy) to it, but not for this reason.

This article is about brainstem (aka hindbrain) vs the rest (midbrain + forebrain).

Evolutionally older animal lineages like fish, reptiles, birds all have midbrain + forebrain as well as the brainstem, so reptilian brain != brainstem.

Where there is some truth to our older "reptilian" brain is that the part of the forebrain called the pallium, that exists even in fish (& reptiles), has much expanded/evolved in mammals giving us our cortex and hippocampus, so you could kinda say that we have a reptilian brain + cortex. Note though that mammals didn't evolve from reptiles - they evolved in parallel from a common ancestor, so maybe better to say we have a fish brain + cortex (& hippocampus).

While the cortex is what gives mammals their intelligence, in birds the pallium evolved in a different way to give them theirs.

am no expert in brain development, still have read about evolution. I like aspects of the way you mentioned this, but seems that putting cortex separately from fish brain isn't quite right. As you mentioned, the cortex is simply a part of the fish brain that evolved much (much, much) larger than a fish's. Maybe:

Fish Brain = brain stem + small cortex (+ other parts of brain)

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 (evolved into)
      |
      V
Human Brain = brain stem + gigantic cortex (+ other parts of brain)

Sorry, know this is splitting hairs, but just thought it a useful distinction to make.

That's not quite right - a fish does not have a cortex.

The pallium of a fish and mammal are structured differently. A fish's (& bird's) pallium has a clustered (aka "nuclear") organization, while a mammal's has a mostly layered organization which is what is referred to as the cortex.

There is also more to a brain than just brainstem + pallium. In a vertebrate it's brainstem (aka hindbrain) + midbrain + forebrain, with the pallium being part of the forebrain.

Ah, I stand corrected:) Maybe a better diagram might be to merge our ideas:

Fish Brain = brain stem + pallium + other parts of brain[1]

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 (evolved into)
      |
      V
Human Brain = brain stem + gigantic cortex(which directly evolved from the pallium) + other parts of brain[1]

Good to know:)

[1] Am using "other parts of the brain" to cover any parts of the brain not covered by the brain stem and cortex