The home country usually refers to the country of permanent residence. It is also usually the country where one was born and/or grew up.
Hospital treatment is inpatient treatment in a hospital. Depending on the type of insurance, inpatient treatment that cannot be postponed in an institution recognised as a hospital is covered under international health insurance (e.g. with the health insurance). Or (for long-term stays abroad) all medically necessary treatments are covered. Please refer to the respective insurance terms and conditions for more information.
The host country is the country in which a person mainly stays during a stay abroad.