This isn't the answer to your question, but perhaps an additional point to consider.

One way humans are rational is when they "reason on paper". We write down most of the initial conditions and then use an algorithm (that's also written down) on them to get to the next step. We write down the result we got. Then we use an algorithm again on this result to get to the next step etc. This is how mathematics and science works. Obviously this is cultural - we learn to do this by learning to write, to do mathematics etc. It's not something ingrained to us.

I think humans can do something similar in their head, but it is greatly limited. The limiting factor is memory - an even remotely complex algorithm will eat up all your memory and you will forget about the results you had already got.

Eg adding 419875 + 87458284 uses the same algorithm as 356 + 472. We can easily do both on paper, but the former is difficult to do in your head, whereas the latter is fairly easy. You give someone an external memory (eg notes) and they both become easy again.

An external memory keeps our reasoning rational.

I think this does answer some of the ambiguity I experienced with these concepts.

Your answer would suggest rational thinking does not come natural to us. That it requires a lot of discipline, external tools and culture.

That would suggest that the rational thinking itself does not take place in the brain, even when humans contribute to it.

The evidence is that a group of humans can extremely reliably solve plenty of different classes of deductive puzzles, with error margins as close to 0 as you can find in the universe. Without that ability modern society wouldn't be possible.

The question is whether deduction is implemented in hardware or in software (that is: is it neurological or cultural).

I'll leave aside the question of deduction for the moment, but it seems quite clear that induction isn't the whole story. At the very least, we have to include abduction (ie. fuzzy pattern matching, or leaping to conclusions) in our repertoire.

To make the case for irrationality, you must make the case using truth, objectivity and reason, and therefore you're implicitly committed to them.

I was pondering an extreme 'sudoku' with the fewest of possible starting numbers: one. Then I realized you could solve it by starting with any solved sudoku and shuffling columns, rows, block-columns, and block-rows, in a manner similar to learning how a Rubik's cube works. That turned it into a vastly simpler deductive problem. I would say that this was creative but not irrational.

Just a small side note, you can solve any such sudoku in the same way you can fill a blank sudoku grid with a valid game state. Take the first 3x3 square and fill it with consecutive numbers 1,2, ..., 9. Then take the next 3x3 square and fill it with 2, 3, ..., 9, 1. Do the same for each next square and you'll end up with a valid board. To solve the one-number-only sudoku you would then simply take the 3x3 square the number is in, fill it with consecutive numbers around the given number, and then fill the rest of the board similar to the above.

Not much different. The advantage with using validity-preserving operations is that you can follow-up with more operations to satisfy additional constraints.

I think what we call rationality is the most likely (aka most active path, when some cost timer in the brain runs out), one can restart it and train it for long periods, but usually its attention is fleeting, because attention is expensive and should be on important things, like food, group, booty and nothing.