I agree that this definition is fuzzy. (So does Carroll, as he makes clear in the text immediately following the bit I quoted.) But no, I don't think it's moving the goalposts, though it may not be putting them where you would prefer them to be.

I take the basic arrow-of-time problem to be something like this: The universe appears to be dramatically asymmetric in time: it is expanding in one time direction and contracting in the other; if we trace its evolution in the direction we call "past" according to our best understanding of the physics, we find a "big bang"; if we go in the direction we call "future" we find a "big freeze". These are distinguished not only by density/scale but also by entropy: the big bang is a much lower-entropy state than the big freeze. Furthermore, we see a similar dramatic asymmetry in our everyday lives: it's easy to break an egg or fry one, not so easy to put it together or turn it raw again. But in the fundamental laws of physics as we currently know them, we find nothing to explain any of this. Weak interactions do indeed show a slight violation of CP-symmetry, hence of T-symmetry, but frying eggs doesn't appear to have much to do with weak interactions; CPT-symmetry would appear to turn our universe into one that "looks just the same" but has time running "the other way"; and nothing in all of this shows any sign of explaining why (the history of) the universe should be so dramatically asymmetric in time.

If weak parity violation really explains anything here, I don't see what. Do you have any grounds for suspecting that weak parity violation explains why we see a very dense low-entropy universe in one direction and a very sparse high-entropy universe in the other? Do you have any grounds for suspecting that weak parity violation explains why smashing an egg is easier than putting it together?

Is there a criterion other than "Sean Carroll thinks so"?

I'm not sure whether this question is really directed at Sean Carroll (complaining that the passage I quoted is vague) or at me (complaining that I'm treating him as some sort of authority). If it's directed at him, the answer he gives is that what you choose to call time-symmetry is up to you and is just a question of terminology, and what really matters is what symmetries the universe actually has. (And, I think he implicitly says, questions about the "arrow of time" remain whatever definitions you choose to adopt.) If it's directed at me, then (1) I endorse his answer and (2) no, I was not using him as an authority, I was using his book as an example of the sort of thing people are usually concerned about in this area.