(a) That's wrong. Performing a specified act is normally sufficient to accept a promise or offer of this type.
But there is another important rule. An act can only operate as acceptance of an offer if the person acting knows of the offer and acts in expectation of receiving what was promised.
If the act is done for an entirely different purpose, or if the promise is only discovered after the act is done, then the act is not treated as acceptance and no enforceable agreement is created. In this example, B did not know of A's promise when he found the buyer for A's house and his act will not be construed as an acceptance of A's offer.