Half siblings

So recently my friend and I have been debating about half siblings. She says that if two children share a mother but have different fathers then they are more related than if they were sharing a father and have two different mothers. I personally think that either way they are related the same amount. My friend says that that's not true because if they share a mother they have more of the same DNA. What are your views on the matter? Do you think I'm wrong or my friend is?