You are saying rational numbers are countable but irrational numbers are not countable (uncountably infinite)

This is true.

The parent is saying that algebraic numbers are countable and transcendental numbers are not countable.

The parent's statement is a level deeper and more remarkable. Another way to say this, is that "almost all" numbers are transcendental.

The parent was not suggesting that his short list of examples represented the extent of transcendental numbers: despite almost all numbers being transcendental, they are individually hard for us to define, since the set is normally defined by excluding sets of numbers that we can easily define.

To clarify: transcendental numbers are a subset/a type of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number If you take out transcendental numbers from the irrationals, those that remain are algebraic and therefore countable. You got it :)

But this is untrue, is it not? Because even if you removed all of the real transcendentals, the real numbers would still be uncountable because of the irrationals.

(Edit: Or are the non-transcendental irrationals countable? I can see how that might be true?)