in the scene (s1ep16) where aomine throws his arm over kagami's shoulders, its 100% meant to be an assertion of superiority. he leans in, gets in kagami's space, looks directly at him and challenges him, taunts him with his words ("entertain me a little"). he's still hoping that kagami will prove to be a challenge but he already has it in his head that nothing will change so he patronizes kagami. there's no extension of friendship or recognition of kagami's skill. he even says he can't believe that this is the person who beat kise and midorima.
kagami's surprised but immediately reacts with anger and throws aomine's arm off him. he's offended and annoyed at aomine's audacity and disrespect for the game. he thinks this will be the game that he'll "wake" aomine up but he's overconfident and takes the loss really hard
but after everything, when kagami sees him at the onsen and he mirrors what aomine had done in the first scene, he pulls aomine into his space. finishes off "the only one who will --" with "-- is us!", directing his words to kuroko but then he looks at aomine and challenges him with his eyes (s2ep37) almost to say "im here to beat you but im also here to be your friend, I'm not going anywhere"
the proximity and the intimacy of that split second challenge throws aomine off - it actually takes him a minute to recover and then it's Kagami who steps aside and takes his arm off aomine's shoulder before aomine has a chance to do it himself
there's something far more personal in kagami's actions in the second scene than aomine's in the first. yes, they truly believe they'll win against him this time, but it's also clear by the time they do beat touou that kagami's intentions weren't only to beat aomine because he was the "one to beat" as the miracles' ace (s2ep43)
some can argue that he did it as a promise to kuroko and maybe that's what it was at first -- but i think it became kagamis personal mission to "wake" aomine up because he can on some level relate to what aomine is feeling - to lose the love for basketball - and so he wants to give that back to him (s1ep15)
and he does -- aomine is startled by kagami's declaration and again, it takes him a minute to recover -- like he can't possibly believe someone would want anything to do with him in that capacity. and he's so quick to accept kagami's rivalry (friendship) because even though aomine keeps his distance, he so easily accepts people? teaches them and roots for them because all he needed was someone to believe in him differently -- to view his skills as amazing instead of monsterous. to recognize him as a person instead of a winning ticket (s2ep43)
from the moment kagami lost to aomine and heard the disappointment in his voice, kagami not only made it his mission to beat him but to extend a rivalry and even a friendship because he knows how lonely it can be at the top (re: himuro). kagami has always been the one ready to accept aomine -- he's the only one to really say with his actions, "ive seen the monster you can be and i embrace it - i embrace you and i'm still here."
