The correct version of this statement is "your belief over the beliefs that you will have after performing a test must be equivalent to your current belief", which seems to be a trivial claim.

It may seem trivial but then again so does the claim that P(A and B) <= P(A), and still...

In particular, I've sometimes caught myself simultaneously having aliefs like ‘if she flees, then she must be a witch’, ‘if she stays, then she must be a witch’, and ‘she may or may not be a witch, and I can't know until I see whether she flees or stays’, and until I read the post about conservation of expected evidence I never realized there was something wrong with that.

The statement, "You should expect that, on average, a test will leave your beliefs unchanged," means that you cannot expect an unbiased test to change you beliefs in a particular direction, as is clear from the original post.

Of course you expect to hold different beliefs after the test. If you didn't, the test would not be worth doing. But you are not more likely to end up at (100% heads, 0% tails) than (0% heads, 100% tails).

On the other hand, if you think it is more likely that you will end up at, say, (0% heads, 100% tails), then you cannot rightly claim that you currently believe the coin to be fair (your 50%, 50% estimate does not reflect your true expectations).

I agree that the statement is not crystal clear. It makes it possible to confuse the (change in the average) with the (average of the change).

Mathematically speaking, we represent our beliefs as a probability distribution on the possible outcomes, and change it upon seeing the result of a test (possibly for every outcome). The statement is that “if we average the possible posterior probability distributions weighted by how likely they are, we will end up with our original probability distribution.”

If that were not the case, it would imply that we were failing to make use of all of the prior information we have in our original distribution.

A misunderstood reading of the statement is that “the average of the absolute change in the probability distribution on measurement is zero.” This is not the case, as you rightly point out. It would imply that we expect the test to yield no information.