Hi there,
Personal liability insurance generally covers an individual or a family rather than a specific property. However, you are required to live with your family at least part-time in order to remain covered by the family policy. You also need to remain unemployed and single. If you move into your own place permanently, get a job or start a family, you will have to get your own personal liability insurance policy.
Household insurance is usually limited to a specific property - in this case your family's residence. However, if the household insurance policy includes coverage for simple theft away from home, your things will still be covered against theft outside of your family's residence.
From the sound of it, you are permanently moving into your own home. If that is the case, you should take out your own household and liability insurance policies to ensure optimum financial protection.
Best regards from Moneyguru
More on this topic:
Personal liability insurance comparison
Household insurance comparison