This word would include me, my brothers and sister as well as my mother and father. It would not include my wife or kids. And as a bonus, is there a word for the group that would only include myself, my wife and kids?
For the term for "group that would only include myself, my wife and kids," I disagree with nuclear family in this case, because that describes a type of family.
The clarification in comments added "I am trying to have someone fill out a form that is relaxed and conversational," and a form would not say "Give the names of your nuclear family". While nuclear here has a specific meaning ("forming a nucleus") it's entirely possible that it will be misunderstood, if it's understood at all in this context.
For this context, immediate family is to be preferred, I feel.
This term appears in OED describing what a "first-degree relative" is.
Designating a member of a person's immediate family, as a parent, sibling, or child. Chiefly in first degree relative.
The term itself is not defined in OED, although immediate is:
Said of a person or thing in its relation to another: That has no intermediary or intervening member, medium, or agent; that is in actual contact or direct personal relation.
While that definition might be taken to include parents or siblings, in normal usage it includes only those who are living together in a household.
And actually, the word household might also suffice.
As for your second request:
a family consisting of two parents and their children, but not including aunts, uncles, grandparents, etc.