Bat-branding
It was Dick who started using the "bat" prefix: the Batcave, Batmobile, Batplane. Bruce supposed that was at least partially his fault for either stamping his symbol on everything or outright shaping it like a bat, but it was a little over the top, even for him. And it was catching. It spread first to Gordon ("Bat-signal") and the League, then the Teen Titans, then Jason and successors—eventually all of Gotham. Now he assigned "bat-missions" and gave "bat-talks" in "bat-speak," "bat-grunts," and "batglares."
Those outside the family were even referring to them collectively as Bats or the "bat clan."
I did try to include Bat-cow, but there just weren't enough words. It cracks me up that the ultra-serious Damian has a pet cow named "Bat-cow" or that Tries-To-Be-Dark-And-Gritty Bruce has a cow in his super-serious crime-fighting cave at all; Damian probably thinks the "Bat" thing is a mark of respect and something of a sign of ownership, and Bruce is just tired and exasperated and picking his battles.