Chi-squared test of independence: is a type II error by itself a bad thing? Let us assume I sell a product that I have modified recently and I want to check if the old or the new version sells better. Therefore, I conduct an experiment in which a hundred people are offered the old version (group A) while another hundred people are offered the new one (group B). In the end, I have four numbers: the number of buyers and non-buyers in A and the number of buyers and non-buyers in B.
Next, I run a chi-squared test of independence to check whether the difference (given there is one) in the proportion of buyers in A and B is statistically significant. In this example, committing a type II error would imply thinking that the old and the new version sell equally well although in reality they do not, which could either mean


*

*(C1) that the old version sells better 

*(C2) or that the new one sells better.


Given that I will stick to the old version, if the observed difference is not statistically significant, the latter case (C2) would be bad for me, because I would give away profit. The former case (C1), on the other hand, would not be bad for me as I would stick to the better performing version of my product. Am I right up to here?
Now, let us assume I perform a power analysis in advance of my experiment which tells me that my test will have a power of 80%, i. e., the probability of committing a type II error will be 20%. However, from the above reflections I know that in my situation committing a type II error is not by itself a bad thing. Under the assumption that C1 and C2 occur equally often, the probability of committing a type II error that is harmful to my business will only be 10%. Is this conclusion correct?
 A: To your first question about whether you would consider C1 and C2 bad outcomes: C1 might not cause you to commit a harmful action (sell the less profitable of two available products), but it does deprive you of valuable information. Suppose that all your competitors start selling the new version, because their faulty experiments convinced them the new version will be more profitable. Your shareholders pressure you to switch to the new version as well. In the absence of strong evidence, you don't have a good reason to refuse. This harms your business.
The key assumption you're making is that you are only relying on this one test to guide your actions. But that's almost never the case. We rely on a continuous stream of new information--such as experiment results--to shape our beliefs. For instance, if we ran an experiment in which (all else equal) smokers did not have significantly higher lung cancer rates than non-smokers, we'd suspect that we'd done something wrong. Why? Because additional information, like the outcomes of similar experiments and medical theory, suggests that this isn't a reasonable conclusion. I think it's risky to conclude that there's no downside risk from C1. 
A: The $\chi^2$-test would not be the best test for your case. 
See the image below for the probability of rejecting the $\chi^2$-test at an $\alpha$ level of 0.01 ($\chi^2 \geq 6.635$) as a function of the probabilities that the old and new products are being bought: $p_{old}$ and $p_{new}$. (with $n=50$ in both groups)

The probability of rejecting the hypothesis $p_{old}=p_{new}$ is symmetric and, indeed, as you identified yourself, does not differentiate between $p_{old}>p_{new}$ or $p_{old}<p_{new}$

You would be better of with a t-test. Comparing the difference between the numbers of sold items for the old and new products. 
Or something similar And then you would wish to set the boundary (to choose either the old or new product) by expressing the costs and profits and the probabilities for these based on the estimates from the observed sales in the hundred cases. (thus, you do not simply choose based on whichever is the largest sold number in your comparison test)
A: As the other answers point out this a very mixed question. 
You have to get the initial question right, otherwise everything that follows will make no sense -- or you will become confused in the interpretation.
First of all determine the real question behind your hypothesis testing. It doesn't matter what method you use to determine some values. And it even doesn't matter if you use the usual terminology of error types. As long as it doesn't mean anything on the "prior" side, you can't expect "meaning" to appear on the "posterior" side of things.
So, figure out what you want: minimize alpha? minimize beta? Have the sharpest decision between one of these? Describe as accurate as possible what kind of error you don't want to make and how to determine as good as possible that it is at most of a certain size. (Keywords: statistical specificity and sensitivity)
The classification "a bad thing" (regardless of which error type) is not useful. Consider the usual examples: e.g. justice: is someone innocent in front of a court? Pharmaceutics: is the drug working? Machinery: is the fault rate acceptable? 
