is there a word or phrase for the family unit that only includes "immediate family" but not spouses and children?
Solution 1:
As for your second request:
Nuclear family:
a family consisting of two parents and their children, but not including aunts, uncles, grandparents, etc.
(Cambridge Dictionary)
Solution 2:
I agree with @FumbleFingers. It can be birth family.
Birth family is normally used to distinguish from adoptive family, but it refers specifically to parents and siblings (excluding wife and kids).