Doubt in calculation of critical region in two-tailed test Consider the following question:-
A tetrahedral die is manufactured with numbers 1, 2, 3 and 4 on its faces. The
manufacturer claims that the die is fair. All dice are tested by rolling 30 times and recording the number of times a ‘four’ is scored. Using a 5% significance level, find the critical region for a two-tailed test that the probability of a ‘four’ is 1/4.
In my textbook, the answer to this question is given as:-

However, this seems incorrect to me as I feel there is no concept of "closest value" in  a critical region as the answer uses. Thus, I feel the correct answer should be $ X≤2$ or $X≥13$.
Who is correct? If the textbook is correct, then why so? Is there a disparity between methods different people use?

When the distribution of a test statistic is discrete, you're left with either a type I error rate lower or higher than your chosen one. It may be that the "lower" option may be much lower and the higher option may only be a little higher.
If the desired significance $\alpha$ was originally an arbitrary choice it may seem strange to avoid going even a little over it. Better to choose whether that's acceptable as appropriate to the situation.
While conservatism is very common, it is not universal. Many choices of actual significance level in the presence of discreteness and asymmetry may lead you to sometimes choose some anti-conservative significance level. Whether this will be accepted by some audience depends on that audience.
The choice of taking the nearest will tend to give you close to your desired significance level on average, but may go over; conservatism won't go over but may sometimes be very far from what you want (with consequences for power). You would need to consider how important in your specific situation it might be not to exceed the nominal significance level (sometimes it might matter a lot, in other circumstances, hardly at all).
In the indicated problem, simply use the convention it tells you to; you can always figure out what the answer would be under some other convention if you wish, but there's no harm whatever in answering such a question on its own terms.
In this case you're either left with a $3.22\%$ test or a $5.90\%$ test; either testing at $64\%$ of the desired significance level or $118\%$ of it. I wouldn't necessarily see an inherent problem with choosing it to be $5.9\%$.
