That little tidbit of information was almost too much for Peter to properly compute, it knocked him off his stride for a moment and all he could do was look between them with his eyebrows raised until he was offered the water. He accepted it with a quiet thanks, visibly relaxing a little.
"Sorry, Dick said it was like a family business, but I didn't want to assume. I- I've got a suit too. No tights or leather, it's a special kind of fabric Mr. Stark made for me. No cape either though. Spider-suit, I'm Spider-Man. Uh, yeah. But that's like really cool. You guys, fighting crime together and stuff."
It really was a wonder that Peter wasn't facepalming. He tried, he really did, he just hadn't dialed in how 'him' he could be around Barbara but when he did it would certainly make him feel less awkward.