(a) That's right. In this case, both A and B have made a mistake about which coat is being bought and sold. They both think it is the coat with the stain, whereas in fact it is one of the unstained coats. This is a 'common' mistake as to the identity of the thing bought and sold.
Mistakes like this do not necessarily prevent the creation of a valid contract. To decide whether the contract is valid despite a common error, the courts ask whether it can be inferred from the known facts that the agreement was conditional on the truth of the mistaken belief. That is: would the parties both still have entered the contract if they had known the truth instead of being mistaken? If they would not, then the agreement was conditional on the mistaken belief, and it is voidable.
In the example, it is obvious from the circumstances that the sale of the coat at the reduced price was conditional on the coat sold being the one with the stain, as the parties believed it was. In view of this, the court will set the contract aside as void. When the contract is declared void, performance must be reversed, as A wants. The result is different when the court finds that the agreement was not objectively conditional on the truth of what the parties believed. In such cases the contract remains valid despite the common mistake.