> the standard scientific protocol tells us to conclude that we have no “significant” evidence that the dice are biased. (Notice that this is subtly different from having evidence that the dice are not biased! Confusing these two statements is a common mistake, even for trained phd scientists, and especially for medical doctors.)

but shouldn't it be "no significant evidence that the dice aren't unbiased"? that's different from either

Yes. Leaving “significant” out of the second one leaves the impression the author is pointing out that non-significant evidence is still evidence. That distracts from the actual point that “absence of evidence is not evidence of absence”