Pretty much so as you said. House is the physical thing. Household is the socioeconomical term for a group of people living under the same roof. "Running a household" for example means taking care of the daily, monthly and yearly things that needs to be done in a home. The word can have slightly different meanings in other contexts, but that gets you pretty far. The Wikipedia-article for Household is also pretty clear.
An owner is in charge of a business. The head of the family is in charge of the household.
The term household has a flavor of responsibility. It's not just about budget/ money it's about how your family presents itself in society, what choices the family makes, what its values are, it's overall character and so on. In some parts of the world it's an old concept that has diminished over time. In some parts of the world where the family structure hasn't changed much from a few hundred years ago it may still have the same meaning.
This is more curiosity than anything else, but I love these conceptual words that translate directly from English to German (and vice versa). In German "halten" is "to hold", so even though you are not literally "holding" a house, the translation is still one-to-one. It exposes their similar roots.
I can't think of many examples now, but similar ones are "ueberblick" - "overview" or rainbow - "Regenbogen" :)
I have listened to the pronunciation of Haushalt on Forvo and dict.cc, as I do with all new words I learn, but I still can't get the "al" sound right. Does anyone have any tips? The end of the word sounds like the "ite" in "bite" to me, but I am sure that isn't right. I don't hear an "l" at all.
I wouldn't really translate "household" as 'home', but when you say the above sentence it probably refers to how well arranged and clean a 'home' or household is, so I guess it's okay.
In general, "Haushalt" can refer both to the household as in chores to do at home, as well as the budget for a company/state, etc. or the consumption of something (e.g. the water household).
http://en.pons.com/translate?q=Wasserhaushaltl=deenin=lf=de Water balance? Hydrologic balance? Are these environmentalist terms?
What does this have to do with a household? LOL Apparently Haushalt can also mean "balance" but not all sources are including that definition.
Sure you can translate the sentence "This is his household" but it would sound like saying "He is the man of the house" which might not come off very well. I really don't hear it used as much any more. Household can mean the physical property of the house and or running of the house and family. I'm not sure what the mean suggests in German but for they meaning explained. I would say for property: That's his house. I would say for running or caring for the house: That's his home. I would expect the person to be conservative and the "man of the house" if he refers to his home as a household.