Instead of glaring at each other stewing in misunderstandings, running around trying to hide stuff to make the misunderstanding last longer, we have proper communication, again. This. This is quality writing. I love it when characters sit down and talk like sane people. If this manga keeps being like this, it's going to find itself a place in my personal top 10 favorites.
Even I understand where Subaru is coming from, she's afraid the school's prejudiced teachers/admins might push to rescind Kaoruko's scholarship over her talking to a guy from a "bottom-feeder" school. I get why her first question is, "Are you her boyfriend?" - because assuming the Kikyo school staff are that bad, then yes, a Kikyo student in a relationship with a Chidori student would sound very improper based on their prejudiced standards. I'm getting a little bit of Romeo and Juliet/star-crossed lovers vibes for this story. But I'm sincerely hoping for a non-tragic ending when the time comes.
I like how our guy Rintaro stood his ground here. If he accepted Subaru's request he'd be silently betraying Kaoruko's trust in him. From Kaoruko's perspective she'd suddenly be ghosted by this kind boy she just met and took a liking to. Subaru is being overprotective here. Her friend can make her own decisions for herself and she's free to be close with whoever she wants to.