Damian's the one who has been--along with Alfred--up to now, reminding us that he's the genetic child. Now we see Dick casually mentioning it in a way that seems out of character for Dick. I could see him saying 'Bruce' or 'Batman' there, but here it feels as odd to me as if one of my siblings called our father "Your Dad".
I disagree and feel that Dick IS downplaying his own connection to Bruce.
I really think it's because this story is the story of Damian, and how, because he's the 'True Son' of Bruce, he's the real inheritor to the throne. That because he's not adopted, he's better than Dick or Tim. And Dick is acknowledging it for us readers.
Also, Damian isn't being insecure here, and doesn't seem to need any reassurance that he is, in fact, Bruce's child. I honestly don't think we're seeing it for any other reason than to drive the point home that Damian is the Real Son, even in Dick's mind.