A hard-headed person looks at what is real — money, time, and what will actually work — and they can make a tough choice without letting liking someone, or feeling sorry, change the decision. In business this is often praise: a hard-headed plan, a hard-headed boss. It is a compliment about being practical, not about being kind. Nothing here is about a head that feels hard when you tap it. The picture is that the head stays in charge, not the heart. Though actually, some speakers also use hard-headed to mean stubborn: the person has already decided, and new facts bounce off them. That second face is messy, and tone decides which one you hear. The sense to learn first is the practical one.