The second one is easier to acknowledge: These ARE my own children. And they're not imaginary. They're very much real.
The phrase you're looking for is biological children. You'd rather have your own biological children than to adopt. THAT is fine. I never pressure anyone to adopt or foster. Its every family's choice. When people act like these are not my children it just bothers me. I'm the one who has to clean up, feed, clothe, parent, etc.
Of course these are my "own kids" even if they're not biological.