I would call this the "Group-inclusion genitive", and it would be parsed as marking that the possessor is a member of the group (or possibly also vice versa, but the opposite direction of inclusion does not interest me for the topic of this particular post.)
I shall use .GiG as an abbreviation for this case.
my.GiG family → my family / the family I am inNow, let's consider a further extension of this: we could maybe combine this with the first person plural pronoun as follows:
your.GiG village → your village / the village you live in
my.GiG us → exclusive 1plalternatively
your.GiG us → inclusive 1pl
? our.GiG us → inclusive 1plIt is my firm bet that blogger's html engine will make the examples above collapse into fewer lines than they should cover despite there being explicit html line breaks in the html source for this post. Here's to hoping against hope that it doesn't.
Of course, such a use of possessive pronouns and personal pronouns could work out even without a particular 'group inclusion genitive' existing, but here, one idea just inspired another.
This could also serve to reduce the need to distinguish, say, colleagues from employees: my.GiG workers = (me and) my colleagues, my workers = my employees.
An interesting thing could be not having the .GiG imply that the group membership extends to the particular statement, e.g. "my.GiG workers are kind" would not necessarily mean that I too am, just that the other members of the group to which I belong are. This could of course maybe be distinguished by means of congruence? If the first person is included, first person congruence is required, if the first person is excluded from the statement, second or third person congruence is required. (And then of course, this distinction will fail whenever in a position where no congruence is available on the verb, so maybe this would just be, say, distinguished for subjects and objects.)