My Tia > Yours
But yeah, there ain't much underlying subtext beneath Rin's denial of being friends with Shiemi - we see this kind of typical "misunderstanding" in anime all the time. Given that Rin appears to be especially hot-headed and arguably bratty even by shonen male lead standards, scenes that amount to "Act first, think
never later" are something i expect to encounter on a fairly regular basis.
I wouldn't blame him too much though in this situation - while definitely a response born out of embarrassment, his interactions with Shiemi haven't been that numerous. Unless i am missing something, beyond his involvement in episode 4 they haven't really interacted much, besides the occasional short exchanges during classroom. Given his personality, i am not surprised his trying to weasel out of a situation he likely considers embarrassing from 'cool guys' perspective (Friends with a girly girl ? Yuck!), and they don't know each other well enough for him to put his pride aside.
So .. yeah, i think he was just being childish