Is there any lingual way to differentiate between the family that I’m a child of and the family that I am a parent of? (I.e., the first family consists of my parents, my siblings, and me; the second one consists of my spouse, my children and me.)
@JAM is right about 'family of origin.' (It does not mean one's 'original family', as contrasted with one's 'new' family.) It does have a clinical feel, though.
"Me and my brood" is a slangy way to refer exclusively to the second group: you and your spouse and children. It's interesting that the spouse is implicitly included (a quick google search will verify this.
From a lexical standpoint, I can't think of anything else.
From a pragmatic standpoint, I would usually go for adverbial or metaphorical expressions in the kinds of cases when the constrasting expressions you are looking for seem necessary. For example, "Under my parent's roof, a kid couldn't sleep over on weekends; under our roof, the kids won't stay at home on weeknights."
Finally, I think you used the word 'lingual' when you mean 'lexical'.
There is a commonly used term, nuclear family that means
a couple and their dependent children, regarded as a basic social unit.
Depending on the characteristics of the speaker, nuclear family might mean different things.
If the speaker were a person who did not have a significant other (spouse, domestic partner, baby mama or papa, etc.) and had no offspring, a reference to nuclear family would probably indicate his or her siblings and parents. However, if such a person were an adult living alone, she or he probably would not use the phrase, since it is most commonly used in reference to a familial group that usually lives together.
If the speaker had a cohabiting significant other, with or without offspring, nuclear family would most likely refer to them collectively (unless the speaker were reflecting to earlier, pre-affiliation and gestational times). However, some people would not describe cohabiting adults as a family unless there were some societally endorsed affiliation.
Finally, I believe that most people would consider a widow and her children (or widower and kids) a nuclear family. Many would consider a divorced, custodial parent and children to be nuclear, although that may vary based on social views and purpose of discussion.
In contrast, there is the concept of extended family that means
a family which extends beyond the nuclear family to include grandparents and other relatives.
This obviously can be much, much broader and is generationally indeterminate, encompassing all of the relatives listed in the question.
Having said all this, I am not aware of unambiguous terms that distinguish, without more context, between me, my sibs and parents and me, my squeeze and offspring.