You say each country defines residence differently. How does the US define it?

Under U.S. immigration law, permanent residence is defined by domicile. Domicile is a legal term that might best be described as where your "home" is as opposed to where your "houses" might be. For example, a person might have a residence in Paris, London and New York, but only one domicile - that is, a place he or she regards as "home". Domicile, under American law, is an ingredient that connotes long term attachment to a place of residence.