I'm GenX (born in '67), and while I haven't had to take care of my parents, I can certainly see why lots of my peers have had that experience. Mom died of cancer 5 years ago but had worried that she might fall victim to Alzheimer's like her mom did. Dad is 77 and in good health, but has just enough going on that my brother and I both have "what if" conversations, especially since he and my stepmother now have custody of one of my nephews. My in-laws don't live with us or require financial support, but my husband is over there at least once a week doing a repair or errand for them.And my own kid? She's 7. We still have lots of childrearing to do.
As for what defines us, I'd say it's the grown up latch key kid mentality: Much more independence than the generations had after us, navigating career environment after job hopping became common and necessary, weaker unions, deregulation, a lighter safety net than the sixties and seventies, and do it yourself retirement planning.