I have never heard of such a subtle difference between "I have got a family" and "I have a family" which seem quite interchangeable to me. Today I learnt that these two sentences have different meanings ("have got" is considered to be present perfect in this use):
I have got a family. = I have recently acquired a family (=my family is young).
I have a car. = I bought my car long go and have been using it ever since.
I have got a car. = I have recently bought a car.
I have got a cat. Do you see these fresh scratches on my hands?
This difference can be critical for translation so I would like to know if it is really crucial and these two sentences don't have the same meaning.