(b) That's wrong. This is very similar to the previous example, but there are important differences. Now A needs the money that B owes him in a hurry. Although he is legally entitled to the overdue rent, he could not enforce that right without considerable delay. So in these circumstances he is getting something additional to what he is already entitled to - the benefit of quick payment and the avoidance of having to take legal action to get the overdue rent, or getting a bank loan to tide him over.
In such cases, as long as B has not improperly or unfairly forced A to make the promise, the practical benefit to A (or the detriment to B in finding the money to pay quickly) will likely be sufficient consideration.